Chaos and Complex Systems: Proceedings of the 5th International Interdisciplinary Chaos Symposium (Springer Proceedings in Complexity) 3030354407, 9783030354404

Table of contents :
Committees
General Chair
Technical Program Committee Co-Chairs
Special Session Committee Co-Chairs
Scientific Committee
Preface
Contents
Contributors
Nonlinear Dynamics and Timeseries Analysis
1 Determinism Testing of Low-Dimensional Signals Embedded in High-Dimensional Multivariate Time Series
1.1 Introduction
1.2 Determinism Test
1.3 Approaches to Test for Determinism
1.3.1 Takens’ Delay Embedding
1.3.2 Combinatorial Approach
1.3.3 Projection Approach
1.4 Dynamical Component Analysis
1.5 Application to Simulated Data Sets
1.5.1 Application to Simulated Data Sets with Noise Components
1.5.2 Application to Simulated Data Sets with Noise Components and Additive Noise
1.6 Application to Epileptic EEG Data Sets
1.7 Conclusion
References
2 CML-Tent Model Chaotic Behavior with Respect to the State and Coupling Parameterse
2.1 Introduction
2.2 Approximate Entropy
2.3 Maximal Lyapunov Exponent
2.4 The 0–1 Test for Chaos
2.5 Conclusions
References
3 Prediction of Echo from Noise Signals by Means of Nonlinear Transform of Signal Spectra
3.1 Introduction
3.2 Algorithm
3.3 Experimental Results
3.4 Conclusion
References
4 Fractal Functions and the Dragon's Mountain: A Functional Equations Perspective
4.1 Introduction
References
5 Movement Characteristics of a Model with Circular Equilibrium
5.1 Introduction
5.2 The Model
5.3 Main Results
5.3.1 Phase Diagrams, the Fourier Spectra, and Bifurcation Diagrams
5.3.2 Fast Approximate Entropy
5.3.3 The 0-1 Test for Chaos
5.4 Conclusions
References
Applications of Nonlinear Dynamics
6 Predictability and Entropy of Supercomputer Infrastructure Consumption
6.1 Introduction
6.2 Prediction Methods
6.3 Entropies and Chaos
6.4 Algorithms Comparison
6.5 Numerical Experiments
6.6 Conclusion
References
7 Chaotic Approach Based Feature Extraction to Implement in Gait Analysis
7.1 Introduction
7.1.1 Literature Survey
7.1.2 Theoretical Background
7.2 Experimental Mechanism
7.2.1 Delay Coordinates Method for Healthy and Patients
7.3 Results and Discussion
References
8 Characterization of Cardiac Cell Electrophysiology Model Using Recurrence Plots
8.1 Introduction
8.2 Beeler-Reuter Model
8.3 Main Results
8.3.1 Recurrence Plots
8.3.2 Recurrence Quantification Analysis
8.3.3 Approximate Entropy
8.4 Conclusions
References
9 Effects of Age and Illness to the Complexity of Human Stabilogram
9.1 Introduction
9.2 Materials and Methods
9.2.1 Dataset
9.2.2 Approximate Entropy
9.2.3 Approximate Entropy Calculation
9.3 Statistical Analysis
9.4 Conclusion
References
Nonlinear Circuits
10 Hybrid Memristor-CMOS Based FIR Filter Design
10.1 Introduction
10.2 Design Approach and Device Modelling
10.2.1 Memristor Ratioed Logic Design
10.2.2 Device Modeling
10.3 Finite Impulse Response Filter Design
10.4 Case Study
10.5 Design Verification
10.6 Experimental Results
10.7 Conclusion
References
11 Chaotic Oscillator for LPWAN Communication System
11.1 Introduction
11.2 Mathematical Model of Chaotic Oscillator
11.3 The Influence of System Parameters on the Dynamics of the System
11.4 Analysis of Chaotic Synchronization
11.5 Conclusions
References
12 Effects of Control Non-idealities on the Nonlinear Dynamics of Switching DC-DC Converters
12.1 Introduction
12.2 The Model of the Boost Converter with Delay
12.3 Results of Bifurcation Analysis
12.4 Conclusions
References
Econophysics
13 Complex Network Time Series Analysis of a Macroeconomic Model
13.1 Introduction
13.1.1 The Vosvrda Idealized Macro-economic Model
13.1.2 Complex Network Time Series Analysis
13.2 Computations and Results
13.3 Conclusions
References
14 On Families of Solutions for Meta-Fibonacci Recursions Related to Hofstadter-Conway $10000 Sequence
14.1 Introduction
14.2 On Families of Solutions
14.3 Conclusion
References
15 Chaotic Solutions for Asset Management Complexity
15.1 Introduction
15.2 Research Method
15.3 Conclusion
References

Citation preview

Springer Proceedings in Complexity

Stavros G. Stavrinides Mehmet Ozer   Editors

Chaos and Complex Systems Proceedings of the 5th International Interdisciplinary Chaos Symposium

Springer Proceedings in Complexity

Springer Proceedings in Complexity publishes proceedings from scholarly meetings on all topics relating to the interdisciplinary studies of complex systems science. Springer welcomes book ideas from authors. The series is indexed in Scopus. Proposals must include the following: – – – – –

name, place and date of the scientific meeting a link to the committees (local organization, international advisors etc.) scientific description of the meeting list of invited/plenary speakers an estimate of the planned proceedings book parameters (number of pages/ articles, requested number of bulk copies, submission deadline) Submit your proposals to: [email protected]

More information about this series at http://www.springer.com/series/11637

Stavros G. Stavrinides Mehmet Ozer •

Editors

Chaos and Complex Systems Proceedings of the 5th International Interdisciplinary Chaos Symposium

123

Editors Stavros G. Stavrinides School of Science and Technology International Hellenic University Thessaloniki, Greece

Mehmet Ozer Department of Physics Istanbul Kultur University Istanbul, Turkey

ISSN 2213-8684 ISSN 2213-8692 (electronic) Springer Proceedings in Complexity ISBN 978-3-030-35440-4 ISBN 978-3-030-35441-1 (eBook) https://doi.org/10.1007/978-3-030-35441-1 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our parents

Committees

General Chair Stavros G. Stavrinides, International Hellenic University, Greece

Technical Program Committee Co-Chairs Mehmet Ozer, Istanbul Kultur University, Turkey Michael Hanias, International Hellenic University, Greece Mattia Frasca, University of Catania, Italy

Special Session Committee Co-Chairs Rodrigo Picos, University of Ballearic Islands, Spain George Sirakoulis, Dimokritus University of Thrace, Greece

Scientific Committee Alon Ascoli, TU Dresden, Germany Dalibor Biolek, University of Brno, Chech Republic Hikmet Caglar, Istanbul Kultur University, Turkey Emine Can, Istanbul Medeniyet University, Turkey Fernando Corinto, Politecnico di Torino, Italy Tiziana Di Matteo, King’s College, UCL, UK Luigi Fortuna, University of Catania, Italy

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Committees

Mattia Frasca, University of Catania, Italy Michael Hanias, TEI-EMT, Greece John Kalomiros, TEI-CM, Greece Lykourgos Magafas, TEI-EMT, Greece Guncel Onkal, Maltepe University, Turkey Vasileios Peristeras, International Hellenic University, Greece Viet-Thanh Pham, Hanoi University of Science and Technology, Vietnam Rodrigo Picos, University of Balearic Islands, Spain Ahmed G. Radwan, Cairo University, Egypt George Sirakoulis, Dimokritus University of Thrace, Greece Ronald Tetzlaff, TU Dresden, Germany

Preface

Nonlinear and complex systems, although deterministic, demonstrate a strange behavior, due to widely diverging outcomes, provoked by small variations in initial conditions or system parameters that lead to inobservance of long-term system-behavior prediction. This very abstractive description explains the reason why Chaos Theory, Nonlinear Dynamics, and Complexity are fascinating areas of interdisciplinary scientific research with a wide range of applications. The interdisciplinary nature and ubiquity of complexity and chaos provide scientists with a motivation to pursue general theoretical tools and frameworks. Nonlinearity and complexity give rise to emergent behaviors, producing novel and interesting phenomena ranging from science and engineering to social sciences, finance, and economy. Consequently, Nonlinear Dynamics, Chaos Theory, and the Science of Complexity appear to have a tremendously increasing Interest, nowadays. The aim of all the International, Interdisciplinary Symposia on Chaos and Complex Systems (CCS) is to bring together scientists, engineers, economists, and social scientists, creating a vivid forum, where discussions on the latest insights and results obtained in the area of Complexity, Nonlinear Dynamics, and Chaos Theory, as well as their interdisciplinary applications, take place. The scope of the 5th International Interdisciplinary Symposium on Chaos and Complex Systems, better known as CCS 2019, which took place in Antalya, Turkey, on May 09–12, 2019, has been enriched with variety of contemporary, interdisciplinary topics, encompassing including but not limited to fundamental theory of nonlinear dynamics, networks, circuits, systems, biology, evolution and ecology, fractals and pattern formation, as well as nonlinear time-series analysis, neural networks, sociophysics and econophysics, management complexity and global systems. Already from the 4th International Interdisciplinary Symposium on Chaos and Complex Systems that took place during May 2012, the organizers are publishing the corresponding proceedings with the Springer Complexity series. Faithful to this tradition, the book that you hold in your hands includes some of the papers presented in the present symposium, addressing some of the areas mentioned above. These proceedings, together with the previous ones, aspire to serve as a compact ix

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Preface

reference book on nonlinear systems, catering to research scholars, interested readers, and advanced learners from multidisciplinary areas. We would like to express our thanks to the Symposium’s International Scientific Committee and the Chairs of the Technical Program Committee and the Special Session Committee, as well as to all those who have contributed to this conference, for their support and advice. We are also grateful to Prof. Dr. Ronald Tetzlaff for his support and the wonderful opening lecture. Our thanks are also due to Prof. Tiziana Di Matteo for the plenary talk, her support, and incentive encouraging. Finally, the editors of this tome are grateful to Springer for the quality of this edition. We hope that this conference series, which begun in 2006, will become a strange attractor for researchers. Thessaloniki, Greece Istanbul, Turkey

Stavros G. Stavrinides Mehmet Ozer

Acknowledgements The editors of the CCS2019 Proceedings would like to thank the Plenary Speakers, Prof. Ronald Tetzlaff from TU Dresden and Tiziana di Matteo from University College London. They would also like to thank all the committee members as well as all the reviewers.

Contents

Part I 1

2

3

4

5

Determinism Testing of Low-Dimensional Signals Embedded in High-Dimensional Multivariate Time Series . . . . . . . . . . . . . . . . C. Frühauf, S. Hartmann, B. Seifert and C. Uhl

3

CML-Tent Model Chaotic Behavior with Respect to the State and Coupling Parameterse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marek Lampart and Tomáš Martinovič

15

Prediction of Echo from Noise Signals by Means of Nonlinear Transform of Signal Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Darya E. Apushkinskaya and Evgeny G. Apushkinskiy

29

Fractal Functions and the Dragon’s Mountain: A Functional Equations Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cristina Serpa

37

Movement Characteristics of a Model with Circular Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marek Lampart and Judita Nagyová

45

Part II 6

7

8

Nonlinear Dynamics and Timeseries Analysis

Applications of Nonlinear Dynamics

Predictability and Entropy of Supercomputer Infrastructure Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jiří Tomčala

59

Chaotic Approach Based Feature Extraction to Implement in Gait Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. İkizoğlu and B. Atasoy

67

Characterization of Cardiac Cell Electrophysiology Model Using Recurrence Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radek Halfar

73

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9

Contents

Effects of Age and Illness to the Complexity of Human Stabilogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radek Halfar

Part III

83

Nonlinear Circuits

10 Hybrid Memristor-CMOS Based FIR Filter Design . . . . . . . . . . . . K. Alammari, A. Sasi, M. Ahmadi, A. Ahmadi and M. Saif

91

11 Chaotic Oscillator for LPWAN Communication System . . . . . . . . . 101 A. Litvinenko, A. Aboltins, D. Pikulins and F. Capligins 12 Effects of Control Non-idealities on the Nonlinear Dynamics of Switching DC-DC Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 D. Pikulins, S. Tjukovs and J. Eidaks Part IV

Econophysics

13 Complex Network Time Series Analysis of a Macroeconomic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 I. P. Antoniades, S. G. Stavrinides, M. P. Hanias and L. Magafas 14 On Families of Solutions for Meta-Fibonacci Recursions Related to Hofstadter-Conway $10000 Sequence . . . . . . . . . . . . . . . 149 Altug Alkan and Orhan Ozgur Aybar 15 Chaotic Solutions for Asset Management Complexity . . . . . . . . . . . 159 G. Cziráki

Contributors

A. Aboltins Institute of Radioelectronics, Riga Technical University, Riga, Latvia A. Ahmadi Department of Electrical & Computer Engineering, University of Windsor, Windsor, Canada M. Ahmadi Department of Electrical & Computer Engineering, University of Windsor, Windsor, Canada K. Alammari Department of Electrical & Computer Engineering, University of Windsor, Windsor, Canada Altug Alkan Graduate School of Science and Engineering, Piri Reis University, Istanbul, Turkey I. P. Antoniades Division of Science & Technology, American College of Thessaloniki, Thessaloniki, Greece Darya E. Apushkinskaya Saarland University, Saarbrücken, Germany; Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia Evgeny G. Apushkinskiy Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia B. Atasoy Department of Mechanical Engineering, Pîrî Reis Maritime University, Istanbul, Turkey Orhan Ozgur Aybar Department of Management Information Systems, Piri Reis University, Istanbul, Turkey F. Capligins Institute of Radioelectronics, Riga Technical University, Riga, Latvia G. Cziráki Alexandre Lamfalussy Faculty of Economics, University of Sopron, Sopron, Hungary J. Eidaks Institute of Radioelectronics, Riga Technical University, Riga, Latvia

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C. Frühauf Center for Signal Analysis of Complex Systems, Ansbach University of Applied Sciences, Ansbach, Germany Radek Halfar IT4Innovations, VŠB—Technical University of Ostrava, Ostrava, Czech Republic M. P. Hanias Physics Department, International Hellenic University, Kavala, Greece S. Hartmann Center for Signal Analysis of Complex Systems, Ansbach University of Applied Sciences, Ansbach, Germany S. İkizoğlu Department of Control & Automation Engineering, Istanbul Technical University, Istanbul, Turkey Marek Lampart IT4Innovations, VŠB—Technical University of Ostrava, Ostrava, Czech Republic; Department of Applied Mathematics, VŠB—Technical University of Ostrava, Ostrava, Czech Republic A. Litvinenko Institute of Radioelectronics, Riga Technical University, Riga, Latvia L. Magafas Physics Department, International Hellenic University, Kavala, Greece Tomáš Martinovič IT4Innovations, VŠB—Technical University of Ostrava, Ostrava-Poruba, Czech Republic Judita Nagyová IT4Innovations, VŠB—Technical University of Ostrava, Ostrava, Czech Republic; Department of Applied Mathematics, VŠB—Technical University of Ostrava, Ostrava, Czech Republic D. Pikulins Institute of Radioelectronics, Riga Technical University, Riga, Latvia M. Saif Department of Electrical & Computer Engineering, University of Windsor, Windsor, Canada A. Sasi Department of Electrical & Computer Engineering, University of Windsor, Windsor, Canada B. Seifert Center for Signal Analysis of Complex Systems, Ansbach University of Applied Sciences, Ansbach, Germany Cristina Serpa Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Instituto Superior de Engenharia de Lisboa, Lisbon, Portugal S. G. Stavrinides School of Science and Technology, International Hellenic University, Thessaloniki, Greece

Contributors

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S. Tjukovs Institute of Radioelectronics, Riga Technical University, Riga, Latvia Jiří Tomčala IT4Innovations, VŠB—Technical University of Ostrava, Ostrava, Czech Republic C. Uhl Center for Signal Analysis of Complex Systems, Ansbach University of Applied Sciences, Ansbach, Germany

Part I

Nonlinear Dynamics and Timeseries Analysis

Chapter 1

Determinism Testing of Low-Dimensional Signals Embedded in High-Dimensional Multivariate Time Series C. Frühauf, S. Hartmann, B. Seifert and C. Uhl

Abstract High-dimensional multivariate time series often consist of a lowdimensional deterministic part. To extract this contribution it is crucial to reduce the dimension of the signal using a dimensionality reduction method. There are three possible approaches, which are compared in this paper: Takens’ delay embedding theorem, a combinatorial approach and projection approaches. After dimensionality reduction of the time-series, the Kaplan-Glass determinism test is applied to compare the obtained signals with respect to deterministic behavior. All introduced methods are applied to simulated noisy data and to EEG data during and out of epileptic seizures.

1.1 Introduction The test for determinism introduced by Kaplan and Glass [1] is used to distinguish between deterministic and stochastic dynamics. It can be applied to embedded time series with time delay and to multivariate time series, e.g. epileptic EEG data, to characterize the deterministic dynamics, which is affected by chaos. The measure of the determinism called k is obtained by computing the total average length of all average unit vectors. The unit vectors are generated by each pass of the trajectory through a box. The boxes are created before by coarse graining the phase space. The determinism test is based on the observation that the tangent to trajectory generated by a deterministic system is a function of position in phase space. Therefore, all tangents to the trajectory in a small area of phase space will have similar orientations [1]. Furthermore, if the trajectory is deterministic the value k will be close to 1 and the system could be described by a set of differential equations. In time series with stochastic components, the value will be smaller than 1, cf. [1].

C. Frühauf (B) · S. Hartmann · B. Seifert · C. Uhl Center for Signal Analysis of Complex Systems, Ansbach University of Applied Sciences, Ansbach, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_1

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Suppose the dimension M of the underlying deterministic dynamics is smaller than the number of channels N of a multivariate signal qi (t) (with i = 1, …, N). Then the questions arises how to calculate κ. Several approaches are possible and are investigated in this paper: (1) For each channel the trajectory in phase space can be reconstructed by Takens’ [2] delay embedding theorem and the Kaplan test can be applied to each channel. The results can be averaged or the maximum can be considered. (2) One or all possible selections of M out of N channels can be chosen and κ can be calculated and averaged over all combinations. (3) Subspaces can be estimated and the projection of the trajectory onto this subspace can be considered for the determinism test. We present results of the Kaplan test of projected signals obtained by the well-known decompositions obtained by principal component analysis (PCA) [3], independent component analysis (ICA) [4] and a recently proposed method, dynamical component analysis (DyCA), which was introduced [5] by our group. These different approaches to calculate κ are applied to simulated noisy data and to electroencephalographic (EEG) data during and out of epileptic seizures. The structure of the paper is as follows. Firstly, the determinism test is recalled. The possible approaches how to calculate the determinism κ are presented in the next section. The recently proposed dimensionality reduction method DyCA is introduced in Sect. 1.4. The applications of the proposed approaches to high-dimensional simulated data sets, based on Lorenz and Rössler system, are presented in Sect. 1.5. In Sect. 1.6, EEG data, during and out of epileptic seizures, are investigated by the presented methods.

1.2 Determinism Test We recall the determinism test of Kaplan and Glass [1]. It works as follows. Firstly, the phase space is coarse grained into equally sized boxes. For each pass i of the trajectory through the kth box a unit vector e i is calculated. Its direction is defined by the two data points, where the trajectory enters and leaves the box. Then the average vector V k of all passes is calculated via Vk =

Pk 1  e i , for Pk > 1, Pk i=1

(1.1)

where Pk is the number of passes through the th box. Finally, the measure of determinism is defined as the average of the lengths of the average vectors V k of all B boxes κ=

B 1  |Vk |. B k=1

(1.2)

1 Determinism Testing of Low-Dimensional Signals …

5

The measure will be close to 1, if the trajectory is deterministic. In time series with stochastic noise the value will be smaller than 1 [1].

1.3 Approaches to Test for Determinism Now we want to propose several approaches to test for determinism if the supposed dimension M of the underlying deterministic dynamics is smaller than the number of channels N of the multivariate signal.

1.3.1 Takens’ Delay Embedding The first method is the reconstruction of the trajectory in phase space by Takens’ [2] delay embedding theorem. The single time series has the form {x0 , x1 , x2 , . . . , xi , . . . , xn } and the reconstructed attractor is given by the following vector sequence   p(i) = xi , xi+τ , xi+2τ , . . . , xi+(m−1)τ .

(1.3)

Here, τ and m are the embedding delay and the embedding dimension. The embedding delay can be determined by the mutual information. It is established by Fraser and Swinney [6] and quantifies the amount of information one has about the state xi+τ if knowing the state xi . For the given single time series {x0 , x1 , x2 , . . . , xi , . . . , xn }, one has to identify the minimum (xmin) and the maximum (xmax ) of this time series.  The absolute value of their difference xmax − xmin  has to be divided into j intervals of equal size, where j is a sufficiently large integer. The equation to compute the mutual information is given by I (τ ) = −

j j   h=1 k=1

 Ph,k(τ ) ∗ ln

 Ph,k(τ ) , Ph ∗ Pk

(1.4)

where Ph and Pk represent the probabilities that there is a time value in the hth and kth bins, respectively. Ph,k(τ ) is the joint probability that xi is in bin h and xi+τ is in bin k. It is proposed to choose the first minimum of I (τ ) for the optimal embedding delay. After reconstructing the trajectory by means of (1.3) and (1.4) with the assumption of a 4-dimensional subspace the Kaplan Test can be applied to each channel. The results can be averaged or the maximum can be considered.

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1.3.2 Combinatorial Approach The second approach is that one or all possible selections of M out of N channels can be chosen and the determinism κ can be calculated for each combination. As mentioned before, the results can be averaged or the maximum can be considered. According to the following equation B=

N! (N − M)! ∗ M!

(1.5)

all possible combinations of N channels taken M at a time can be generated.

1.3.3 Projection Approach The last proposed approach is based on estimated subspaces and the projection of the trajectory can be considered for the determinism test. The projected signals can be obtained by the well-known decompositions obtained by principal component analysis (PCA, [3]), independent component analysis (ICA, [4]) and a new method, dynamical component analysis (DyCA, [5]). In the next section, the essential concepts of DyCA and the formulation of the dimensionality reduction method are introduced.

1.4 Dynamical Component Analysis One has given a multivariate time series q(t) ∈ R N . On the basis of the assumption that the given time series has a low-dimensional deterministic dynamics, it can be decomposed into q(t) =

M 

xi (t)w i ,

(1.6)

i=1

where xi (t) is the time-dependent amplitude with vectors w i ∈ R N . Note, that M  N. For the signal decomposition with DyCA, the system has to be describable by a system of ordinary differential equations, which consists of linear differential equations,

1 Determinism Testing of Low-Dimensional Signals …

x˙1 =

M 

7

a1,k xk

k=1

x˙m =

.. . M 

(1.7) am,k xk

k=1

and non-linear differential equations, x˙m+1 = f m+1 (x1 , x2 , . . . , x M ) .. .

(1.8)

x˙ M = f M (x1 , x2 , . . . , x M ),

where f j are non-linear smooth functions. Important is, thereby, that there are more linear than non-linear equations, i.e. m > M2 . It is assumed that the parameters ai,k and the exact form of the functions f are unknown. To get the projection vectors u i , v j ∈ R N of the underlying dynamics, one has to minimize the least square error cost function

2  

T

q˙ u − j a j q T v j

2 t

D(u, v, a) =

2

q˙ T u

2

(1.9)

t

where .t denotes the time average. Denote the correlation matrices of the signal with itself, of the signal with its derivatives, and the signal derivatives with itself by       C0 = qi q Tj t , C1 = q˙i q Tj t and C2 = q˙i q˙tT t .

(1.10)

After rewriting and including the correlation matrices of the signal the minimization yields a generalized eigenvalue problem C1 C0−1 C1T u = λC2 u.

(1.11)

The solution of the generalized eigenvalue problem leads to the projection vectors u i and the eigenvalues λi . Afterwards, the other projection vectors v i can be calculated by v i = C0−1 C1T u i .

(1.12)

The projection of the data is done onto span{u1 , . . . , um , v 1 , . . . , v m } = R N ,

(1.13)

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where m corresponds to the number of eigenvalues λi close to 1. The eigenvalues λi of the generalized eigenvalue problem (1.11) show the quality of the least-square-fit of the linear differential equations (1.7). According to the following equation Dmin = 1 − λ

(1.14)

the value Dmin becomes 0, when the eigenvalues λi get close to 1. This leads to amplitudes xi , which can be modelled by a system of ordinary differential equations.

1.5 Application to Simulated Data Sets The simulated data sets consist of high-dimensional signals obtained by integrating Lorenz and Rössler ODEs and embedding the signals in noisy 20-dimensional spaces. The Lorenz attractor [7] is a strange attractor given by the system of ordinary differential equations X˙ = aY − a X Y˙ = bX − X Z − Y Z˙ = X Y − cZ ,

(1.15)

with a = 10, b = 28 and c = 83 . The Rössler attractor [8] is given by the system of ordinary differential equations X˙ = −(Y + Z ) Y˙ = X + aY Z˙ = b + X Z − cZ ,

(1.16)

with a = 0.15, b = 0.2 and c = 10. The trajectories of these systems are simulated by means of a (4, 5)-Runge-Kutta integration method. In addition to that, we add noise components with a signal to noise ratio of 10 dB. The simulated data sets with noise components are defined by the following equation qi (t) =

3  i=1

xi (t)wi +

17 

r j (t)w j ,

(1.17)

j=1

where xi (t) represent the amplitudes of the Lorenz/Rössler signal, r j (t) the amplitudes of the noise components and w i and w j linear independent randomly chosen vectors. To investigate the limits of our approach, we added additive noise n(t) to

1 Determinism Testing of Low-Dimensional Signals …

9

both simulated data sets in (1.17) and increased the additive noise level to a signal to noise ratio of 15 dB. After the simulation of the introduced data sets the combinatorial and projection approaches are applied to both simulation data sets with the assumption of 3- and 4-dimensional subspaces.

1.5.1 Application to Simulated Data Sets with Noise Components Firstly, after computing all possible combinations of 3 out of 20 and 4 out of 20 channels, the Kaplan Test is applied to each combination of single time series. The results are averaged and the maximum of the determinism κ is considered. Table 1.1 presents the results of the averaged determinism κ with standard deviation and the maximum of κ for all possible combinations. It can be noticed that the average of κ as well the maximum of the value for both simulated data sets are not close to 1. Applying the third proposed technique to both simulated data sets the results presented in Table 1.2 are obtained. It shows a comparison of the results of the determinism test after the projection of the trajectory by PCA, ICA and DyCA. The projection of the trajectory by PCA and ICA yield values κ, which are also smaller than 1. In contrast, the application of DyCA enabled a projection of the relevant subspace. This is approved by the calculated determinism κ, which is close to 1 for both data sets. Table 1.1 Averaged determinism κ with standard deviation and the maximum of κ for all possible combinations of 3 out of 20 channels and 4 out of 20 channels of the two datasets Lorenz

Rössler 3 out of 20

4 out of 20

3 out of 20

4 out 20

Average of κ

0.4236

0.5005

Average of κ

0.5688

0.5874

Standard deviation

0.0406

0.0324

Standard deviation

0.0429

0.0349

Maximum of κ

0.5234

0.6131

Maximum of κ

0.6746

0.6918

Table 1.2 Comparison of the results of the determinism κ for both simulated datasets after application of PCA, ICA and DyCA Lorenz

Rössler 3-dim

4-dim

PCA

0.5873

0.5799

ICA

0.1811

0.5080

DyCA

0.9582

0.9528

3-dim

4-dim

PCA

0.3391

0.3839

ICA

0.0670

0.2819

DyCA

0.9864

0.9916

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C. Frühauf et al.

Table 1.3 Averaged determinism κ with standard deviation and the maximum of κ for all possible combinations of 3 out of 20 channels and 4 out of 20 channels of the signals Lorenz

Rössler 3 out of 20

4 out of 20

Average of κ

0.3909

0.4390

Standard deviation

0.0530

0.0346

Maximum of κ

0.5127

0.5270

3 out of 20

4 out of 20

Average of κ

0.3642

0.4334

Standard deviation

0.0306

0.0335

Maximum of κ

0.5237

0.5954

Table 1.4 Comparison of the results of the determinism κ for both simulated datasets after application of PCA, ICA and DyCA Lorenz

Rössler 3-dim

4-dim

PCA

0.6300

0.6122

ICA

0.6356

0.6810

DyCA

0.4599

0.5661

3-dim

4-dim

PCA

0.3824

0.4030

ICA

0.4349

0.5463

DyCA

0.5743

0.6504

1.5.2 Application to Simulated Data Sets with Noise Components and Additive Noise In this section, the results of the application to simulated data sets with noise components and additive noise are discussed. For large signal to noise ratios we obtain similar results as presented in Sect. 1.5.1. Lowering the signal to noise ratio down to 15 dB yields results of the averaged determinism κ with standard deviation and the maximum of κ for all possible combinations which are presented in Table 1.3. Again the value for the determinism κ is still smaller than 1. We get similar results after the projection of the trajectory by PCA, ICA and DyCA, which are shown in Table 1.4. None of the results indicates deterministic dynamics. Even DyCA cannot project the relevant subspace due to the level of additive noise which corrupts the signal in such a way that a projection does not yield a deterministic signal.

1.6 Application to Epileptic EEG Data Sets We investigated EEG data of patients with petit mal epilepsy (absences). The data was recorded by 25 surface EEG sensors with 256 Hz sampling rate. The different approaches to calculate the determinism κ were applied to three data sets during and out of epileptic seizures with the assumption of a 4-dimensional subspace.

1 Determinism Testing of Low-Dimensional Signals …

11

Table 1.5 Averaged determinism κ with standard deviation and maximum of three datasets for the time series during and out of seizure after delay embedding of each channel Absence 1

Out of absence 1

Absence 2

Out of absence 2

Absence 3

Out of absence 3

Average of κ

0.9273

0.4817

0.9410

0.6054

0.9423

0.5691

Standard deviation

0.0220

0.0667

0.0182

0.0332

0.0184

0.0587

Maximum of κ

0.9522

0.6316

0.9712

0.6504

0.9706

0.6479

Table 1.5 shows the results obtained by technique (1)—attractor reconstruction with time delay coordinates and averaging κ for all channels for three different EEG datasets during and out of epileptic seizures. The values of the determinism κ are close to 1 for the time series during seizure. In contrast, for the time series out of seizure the values of κ are smaller than 1. Figure 1.1 shows the best possible reconstructed phase spaces of the channels of the second data set for the time series during and out of seizure. Table 1.6 presents results of the averaged determinism κ with standard deviation and maximum for all possible combinations of 4 out of 25 channels of the signal. It can be seen, that the values are smaller. Nevertheless, there exists also the contrast, that the determinism κ is much smaller for the time series out of seizure. In Fig. 1.2 the projections of the best combinations of the second data set are shown. A comparison of the results of the determinism test after projection into a 4dimensional subspace with PCA, ICA and DyCA is shown in Table 1.7. It is noticeable, that the determinism κ is close to 1 for the time series during seizure after the projection of the trajectory by DyCA. In contrast, the PCA and ICA projections do not indicate deterministic dynamics. Therefore, DyCA enables the

Fig. 1.1 Attractor reconstruction of the second data set by Takens’ delay embedding theorem. 3D-plot of the embedded channels 8 (left) and 18 (right) with the greatest value of κ. The left figure presents the trajectory of the time series during seizure and the right figure presents the trajectory of the time series out of seizure

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Table 1.6 Averaged determinism κ with standard deviation and maximum of three datasets for the time series during and out of seizure after computing all possible combinations of 4 out of 25 channels Absence 1

Out of absence 1

Absence 2

Out of absence 2

Absence 3

Out of absence 3

Average of κ

0.8058

0.4195

0.8230

0.4616

0.7885

0.4364

Standard deviation

0.0554

0.0424

0.0516

0.0366

0.0518

0.0441

Maximum of κ

0.9200

0.5990

0.9447

0.5862

0.9262

0.5657

Fig. 1.2 3D-plot of the trajectories with the greatest κ value of the second data set. As in Fig. 1.1 the best combination of the EEG channels during seizure is on the left hand side and on the right hand side the best combination of the EEG channels out of seizure is presented

Table 1.7 Comparison of the results of the determinism κ for the time series during and out of seizure after application of PCA, ICA and DyCA Absence 1

Out of absence 1

Absence 2

Out of absence 2

Absence 3

Out of absence 3

PCA

0.7276

0.6866

0.8494

0.4330

0.8428

0.4416

ICA

0.7721

0.5785

0.8352

0.3576

0.748

0.4119

DyCA

0.9617

0.3601

0.9705

0.3351

0.9633

0.3488

best projection of the relevant subspace during seizure. The projected signals can be seen in Figs. 1.3 and 1.4. The trajectory obtained by DyCA projection leads to the best structured trajectories in phase space confirming the values of κ presented in Table 1.7.

1 Determinism Testing of Low-Dimensional Signals …

13

Fig. 1.3 Projected signals obtained by PCA (left) and ICA (right) during seizure

Fig. 1.4 Projected trajectory of DyCA during seizure

1.7 Conclusion For simulated multivariate data sets we show that the projection by DyCA leads to better results with respect to the determinism κ than PCA and ICA projections as well as calculation of κ from single time series and combinations of channels. This is true as long as the level of additive noise does not corrupt the deterministic signal too much. All results concerning the EEG signal of epileptic seizures show that the time series during seizure are “more deterministic” than the out of seizure dynamics assuming a 4-dimensional subspace: the κ values are closer to 1 during seizure. The DyCA projection we proposed in [5] outperforms all other techniques investigated in this study. This is due to the underlying concept of DyCA of searching projections leading to amplitudes, which can be modelled by ODE. In addition to that, the application of DyCA to epileptic EEG data is based on the observation [9] that there exists Shilnikov chaos. That is why the seizure dynamics can be described by ODE, which consists of two linear and one non-linear equations [10]. Figure 1.4 presents a similar homoclinic orbit of Shilnikov chaos by DyCA projection of the trajectory during seizure. Moreover, it is a useful method to get the projection of the trajectory in a more efficient way compared to technique (1.1), which produced similar results.

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Acknowledgements We thank the Epilepsy Centre at the Department of Neurology, Universitätsklinikum Erlangen for provided data and for fruitful ideas and discussions.

References 1. D.T. Kaplan, L. Glass, Direct test for determinism in a time series. Phys. Rev. Lett. 68(4), 427–430 (1992) 2. F. Takens, in Dynamical Systems and Turbulence, Warwick, 1980, ed. by D.A. Rand, L.S. Young (Springer, 1981), p. 366 3. K. Pearson, On lines and planes of closest fit to a system of points in space. The London, Edinburgh, and Dublin Philos. Mag. J. Sci. 6(2), 559–572 (1901) 4. A. Hyvärinen, E. Oja, Independent component analysis: algorithms and applications. Neural Netw. 13(4–5), 411–430 (2000) 5. B. Seifert, K. Korn, S. Hartmann, C. Uhl, Dynamical component analysis (DyCA): dimensionality reduction for high-dimensional deterministic time-series, in Proceedings of the IEEE 28th International Workshop on Machine Learning for Signal Processing (MLSP 2018) 6. A.M. Fraser, H.L. Swinney, Independent coordinates for strange attractors from mutual information. Phys. Rev. A 33(2), 1134–1140 (1986) 7. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963) 8. O.E. Rössler, An equation for continuous chaos. Phys. Lett. 57A(5), 397–398 (1976) 9. L. van Veen, D.T.J. Liley, Chaos via Shilnikov’s saddle-node bifurcation in a theory of the electroencephalogram. Phys. Rev. Lett. 97, 208101 (2006) 10. K. Korn, B. Seifert, C. Uhl, Dynamical component analysis (DYCA) and its application on epileptic EEG, ICASSP 2019, in 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Brighton, United Kingdom (2019)

Chapter 2

CML-Tent Model Chaotic Behavior with Respect to the State and Coupling Parameterse Marek Lampart and Tomáš Martinoviˇc

Abstract The main aim of this paper is the study of dynamical properties of the Laplacian-type coupled map lattice induced by the Tent family on a periodic lattice depending on two parameters: the state parameter of the Tent map and the coupling constant. For this purpose, tools like approximate entropy, maximal Lyapunov exponent, and the 0–1 test for chaos are introduced and applied to numerical simulations performed using a supercomputer.

2.1 Introduction Many complex systems rising from applications are studied from both the qualitative and also quantitative, long term dynamics point of view. In the last decades, this approach was successful in considerations on the states evolution in the coupled map lattices (CML), which are a class of models of extended media in which the relations between temporal evolution and spatial translation play a significant role [1, 2]. It has been discovered that CML systems exhibit a variation of space-time patterns, which are ordinary to other spatially enlarged systems (see, e.g., [3–5], or [6] and references therein). There are several concepts of communication between states. A simple, yet interesting, model   consists in one-way coupled logistic lattices, which are modeled as a lattice x i i∈Z ⊂ [−1, 1] with evolution modeled by a discrete dynamical system M. Lampart (B) IT4Innovations, VŠB—Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, Czech Republic e-mail: [email protected] Department of Applied Mathematics, VŠB—Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic T. Martinoviˇc IT4Innovations VŠB—Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_2

15

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M. Lampart and T. Martinoviˇc

(n denotes values of the lattice at ith iteration) i = (1 − ε)g(xni ) + εg(xni−1 ), xn+1

(2.1)

called as Laplacian-type coupling for the particular choice of map g, which is taken from logistic family gμ (x) = 1 − μx 2 for some μ (e.g. see [7, 8]). Some authors prefer to consider logistic family on [0, 1], i.e. f μ (x) = μx(1 − x) where in most considered cases μ = 4 see e.g. [9], or the Tent family Ts (x) = s(1 − |2x − 1|) see, e.g., [5, 6]. Note that the dynamics of all mentioned above interval maps x → 1 − μx 2 , x → μx(1 − x), and x → s(1 − |2x − 1|) is “similar”, in the sense that they share most of interesting dynamical characteristics, see e.g. [10, 11]. The model (2.1) was researched by many authors from different points of view in the last 30 years. Many interesting results were reached; however, most of them were obtained as a result of numerical simulations and much smaller insight was done using analytic methods. Spectral properties of the model (2.1) generalizations were discussed in [12] and conditions for the stability of spatially homogeneous chaotic solutions were presented. Some studies whether there is a synchronization were undertaken in [13] and later, e.g., in [14, 15], or [16] (for more comments, see references therein). The existence of a wavelike solution in the model (2.1) for which the spatiotemporal periodic pattern can be predicted was investigated in [17] (see also references therein). The bifurcation phenomena [18], loss of synchronization [7], periodic structure, multiple attractors, entrained and phase reversed patterns and chaos [19, 20], and the existence of chaos in the sense of Li and Yorke for some range of parameters [21] were also observed. The model (2.1) was also studied for negative coupling constant  revealing the existence of a forward invariant curve [22]. It is very easy to see that the model (2.1) exhibits chaotic behavior for zero coupling constant ε under the condition that g is chaotic since in this case, the model is acting just as a Cartesian product g × g of chaotic g. There is also always an invariant diagonal, where dynamics is the same as g. In this direction a natural question arises, whether large off-diagonal regions of chaotic dynamics could be observed for a nonzero coupling constant. One of our aims is to detect regions outside the diagonal, where chaotic dynamics can be supported. A natural procedure is to divide the region of parameters s ×  into several areas for which regular or irregular movement character is observable. Main aims of this paper are natural continuation of the research started in [9] and further developed in [23] where the investigated model was generated by the logistic family. On contrary, in this paper the Tent family will be applied. The difference between these families were deeply studied in [11] showing the basic distinction that is distribution. The goal of this paper is to use the 0–1 test for chaos, approximate entropy, and the maximal Lyapunov exponent to detect the behavior of dynamical properties of the model (2.3) on the parameters set. This study is focusing on the two dimensional cases (i.e., a lattice with 2-periodic entries), since in a higher dimension (i.e., a larger period on the lattice) the problem could be dealt with similarly by analogous calculations and arguments.

2 CML-Tent Model Chaotic Behavior with Respect … Fig. 2.1 Graph of Ts , Ts2 , and Ts3 and the graph restricted to the core Is (bounded by box) for s = 0.9

17

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Firstly, define the family of Tent maps Ts : [0, 1] → [0, 1], where 0 ≤ s ≤ 1, by Ts (x) = s(1 − |2x − 1|).

(2.2)

The interval Is = [Ts2 (1/2), Ts (1/2)] is called the core of Ts for s ∈ (1/2, 1], see Fig. 2.1. For s ∈ [0, 1/2] the interval Is is still well defined but does not have such nice properties. Namely, I1/2 = {1/2}, I0 = {0} and for s ∈ [0, 1/2) it is not invariant under Ts , i.e. Ts (Is ) ⊂ Is , hence not remarkable in these cases. The core Is is strongly invariant, that is Ts (Is ) = Is , and every point from (0, 1) is attracted to Is . The dynamics on the core can be very rich. In [24] the authors show that for the family of Tent maps Ts the dynamics on the core is topologically exact for some range of parameters, which, generally speaking, means that most rich dynamical behavior is present in the core. Now, by coupled Tent map (with coupling constant ε) the following map of the square F : [0, 1]2 → [0, 1]2 is meant, F(x, y) = (Fx (x, y), Fy (x, y))

(2.3)

Fx (x, y) = (1 − ε)Ts (x) + εTs (y), Fy (x, y) = (1 − ε)Ts (y) + εTs (x).

(2.4) (2.5)

where

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M. Lampart and T. Martinoviˇc

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2.2 (left)—union of all fixed points and (right)—union of all 2-periodic points MAGENTA that are not fixed of the model (2.3).

The first step in the study of dynamical properties of the model (2.3) is to look for the structure of periodic points. It is easy to see, by straight forward computations, that Fix(F) the set of all fixed points contains at most four points if s = 1/2. For example, if s = 1 and ε = 0 it is Fix(F) = {(0, 0), (2/3, 0)(0, 2/3), (2/3, 2/3)}, and if s = 1 and ε = 1 it is Fix(F) = {(0, 0), (4/5, 2/5)(2/5, 4/5), (2/3, 2/3)}. The boundary cases of coupling parameters are summarized in the following two theorems, that can be proved by direct calculations. Theorem 2.1 For ε = 0 it is Fix(F) = {(0, 0), ( ps , 0)(0, ps ), ( ps , ps )} where ps = 2s/(1 + 2s).  Theorem 2.2 For ε = 1 it is Fix(F) = {(0, 0), (qs , 2sqs ), (2sqs , qs ), ( ps , ps )}  where ps = 2s/(1 + 2s) and qs = 2s/(4s 2 + 1). Unfortunately, the general formulae of Fix(F) are hard to get. The union of all fixed points of ∪s∈[0,1] ∪ε∈[0,1] Fix(F) is shown in the left part of Fig. 2.2, where black line stands for ∪ε∈[0,1] Fix(F) assuming s = 0.9 and red line for ∪s∈[0,1] Fix(F) assuming ε = 0.9. The union of all periodic points of period 2 without fixed points is shown in the right pat of Fig. 2.2 by magenta color, fixed points are also shown in green. Some attempts to perform analytic analysis of situations when this occurs were undertaken in [25] by a clever change of parameters. The complexity of the model (2.3) for the specific choice of parameters is depicted in Fig. 2.3, where global attractors and contour plots (two-dimensional kernel density estimation with an axisaligned bivariate normal kernel, evaluated on a square grid) were performed on a grid √ distanced√alongside axes on the intervals √ of 1000 ×√1000 points equidistantly [ 2/2 − 0.7; (2)/2 + 0.29] and [ 3/3 − 0.57; 3/3 + 0.42] for x-axis and yaxis, respectively. Each point was iterated 105 times.

2 CML-Tent Model Chaotic Behavior with Respect …

19

Fig. 2.3 Global attractor (black dots) and contour plot (red lines) of model (2.3) for: upper-left s = 0.71 and ε = 0.92; upper-right—s = 0.75 and ε = 0.75; lower-left—s = 0.75 and ε = 0.85; lower-right—s = 0.85 and ε = 0.85

Note that when ε = 0, then F = Ts × Ts . Therefore, it is natural to expect that for small values of ε dynamics of F will be in some sense similar to the dynamical independence of coordinates Ts × Ts . At least we may hope for two effects. First of all, the invariant subsets outside the diagonal should survive. Second, entropy should increase over the value h top (Ts ) supported on the diagonal. Theorem 2.3 Let s ∈ [0, 1/2) and (x, y) ∈ [0, 1]2 . Then for any ε ∈ [0, 1] it holds lim F n (x, y) = (0, 0).

n→∞

Proof Firstly note, that limn→∞ Tsn (x) = 0 for s ∈ [0, 1/2) and x ∈ [0, 1]; and Tsn (1/2) ≥ Tsn (x) for any n and x ∈ [0, 1/2]. Hence for if F(x, y) ∈ [0, 1/2]2 then F n (x, y) ∈ [0, Tsn (1/2)]2 or any n ending the proof. 

20

M. Lampart and T. Martinoviˇc

So, by the above given Theorem 2.3 the model is showing symmetry with respect to the horizontal line at 1/2, due to the Theorem 2.4 all points of the state space [0, 1]2 are vanishing to (0, 0) while s ∈ [0, 1/2). Hence main results of the paper will be depicted only on the set of parameters [1/2, 1] × [0, 1]. In the following theorem we use the notation ε F of lower left subscript to distinguish between different choices of coupling parameters. Theorem 2.4 Let ε2 = 1 − ε1 where ε1 ∈ [0, 1/2]. Then for any (x, y) ∈ [0, 1]2 it holds 2 2 ε1 F (x, y) = ε2 F (x, y). Proof Pick ε1 ∈ [0, 1/2] then ε1 F(x,

y) = ((1 − ε1 )Ts (x) + ε1 Ts (y), (1 − ε1 )Ts (y) + ε1 Ts (x)) = (ε1 Fx (x, y), ε1 Fy (x, y)).

Now, ε2 F(x,

y) = (ε1 Ts (x) + (1 − ε1 )Ts (y), ε1 Ts (y) + (1 − ε1 )Ts (x)) = (ε1 Fy (x, y), ε1 Fx (x, y)),

and ε2 F

2

(x, y) =

(ε1 Ts (ε1 Fx (x, y)) + (1 − ε1 )Ts (ε1 Fy (x, y)), ε1 Ts (ε1 Fy (x, y)) + (1 − ε1 )Ts (ε1 Fx (x, y))) =

ε1 F

ending the proof.

2

(x, y) 

2.2 Approximate Entropy Approximate entropy is a pretty robust measure of complexity or irregularity. The main advantages are that it can be computed on short time series and it allows to compare the differences in complexity of the same system with different parameters settings, see, e.g., [26, 27], to be detected. To compute the approximate entropy, two parameters must be set: embedding dimension m and neighbourhood threshold r . Let x(t) ∈ R for t = {1, 2, . . . , N } be a time series with N observations. Then embedded vector X (t) at time t, is defined as X (t) = [x(t), x(t + 1), x(t + 2), . . . , x(t + (m − 1))], where t is the observed time and m is the embedding dimension. The maximum distance of embedded vectors is computed as follows:

2 CML-Tent Model Chaotic Behavior with Respect …

D(i, j) = d(X (i), X ( j)) =

max

21

k=0,1,...,m−1

|x(i + k) − x( j + k)|,

(2.6)

for i, j = {1, 2, . . . , N − (m − 1)}. Compute the thresholded version of the distance with threshold given by r :  dr (i, j) =

1, 0,

D(i, j) < r other wise,

(2.7)

for i, j ∈ {1, 2, . . . , N − (m − 1)}. Compute Cim (r ) as a ratio between points in the neighbourhood of i and the number of embedded vectors.  N −(m−1) Cim (r )

=

j=1

dr (i, j)

N − (m − 1)

.

(2.8)

Then compute the average of logarithm of all the Cim (r ) 1  (r ) = N − (m − 1) m

N −(m−1) 

ln Cim (r ).

(2.9)

i=1

Finally, approximate entropy for the finite time series with N data points is computed as (2.10) ApEn(m, r, N ) = m (r ) − m+1 (r ). Pincus [27] suggests to use at least 103 observations for robust estimation. The examination of model (2.3) was done on the 105 iterations of the system. The output of approximate entropy estimation depending on s and  is visible in Fig. 2.4. Here, the green mark stands for regular movement, i.e. approximate entropy is close to zero and the red marks for very complex system behavior.

2.3 Maximal Lyapunov Exponent If two infinitesimally close points of the system diverge exponentially in time and remain in the same compact space, it is considered that the system is chaotic, see, e.g., [28]. The measure of this divergence is called Lyapunov exponent. The positive value of the Maximal Lyapunov exponent (MLE) indicates the system is chaotic, for zero case the bifurcation occurs, and a negative Lyapunov exponent detects regular (periodic) movement. MLE was computed on the 105 iterations of the system from which the first 103 iterations were discarded. The Lyapunov exponents were computed as

22

M. Lampart and T. Martinoviˇc

Fig. 2.4 Approximate entropy of the model (2.3) depending on s and ε parameters

  N  d Fx (x, y)  1  , log   N i=1 x   N  d Fx (x, y)  1  y ,  L Ex = log   N i=1 y   N  d Fy (x, y)  1  x ,  L Ey = log   N i=1 x   N  d Fy (x, y)  1  y ,  L Ey = log   N i=1 y

L E xx =

(2.11)

(2.12)

(2.13)

(2.14)

where N is the number of observations of the input time series. Then the maximal Lyapunov exponent was taken as MLE. Results of the MLE for different settings of s and  are shown in Fig. 2.5.

2.4 The 0–1 Test for Chaos The 0–1 test for chaos was introduced in [29] to distinguish between regular and chaotic dynamics in deterministic dynamical systems. As an output of the test, 0 stands for regular movement and 1 for chaotic patterns. As opposed to the computational methods of the Lyapunov exponent, this method is direct on tested data, i.e. no preprocessing and only minimal computational effort is required. This method

2 CML-Tent Model Chaotic Behavior with Respect …

23

Fig. 2.5 The maximal Lyapunov exponent of the model (2.3) depending on s and ε parameters

was originally stated as regression one, and later on in [30] it was improved as correlation that is faster and qualitatively gives better results; it is faster in terms of convergence. This correlation method works for a given set of observations φ( j) for j ∈ {1, 2, 3, . . . N } as follows. Firstly, compute the translation variables for suitable choice of c ∈ (0, 2π ): pc (n) =

N 

φ( j) cos( jc), qc (n) =

j=1

N 

φ( j) sin( jc),

j=1

then the mean square displacement Mc (n) = lim

N →∞

N 1  [ pc ( j + n) − pc ( j)]2 + [qc ( j + n) − qc ( j)]2 N j=1

where the limit is confident by calculating Mc (n) only for n ≤ n cut where n cut N , and put n cut = N /10. Now, let us estimate the modified mean square displacement ⎛

⎞2 N  1 1 − cos(nc) Dc (n) = Mc (n) − ⎝ lim . φ( j)⎠ N →∞ N 1 − cos(c) j=1 Put ξ = (1, 2, . . . n cut ),  = (Dc (1), Dc (2), . . . Dc (n cut )). Finally, we get the output of the 0–1 test as the correlation coefficient of ξ and  for fixed parameter c K c = corr(ξ, ) ∈ [−1, 1].

(2.15)

24

M. Lampart and T. Martinoviˇc

Fig. 2.6 The 0–1 test for chaos of the model (2.3) depending on s and ε parameters

Obviously, K c is dependent on the choice of c, and as it was pointed out in [30], it is enough to get K as the output of the 0–1 test, as the limiting value of all K c . Our tests confirm experience of [30] that it is sufficient to introduce K = median(K c ). To avoid the resonances distorting the statistics, parameter c is chosen from the restricted interval (π/5, 4π/5) for all computations, see [30]. In our tests, 100 equidistant samples from π/5 to 4π/5 were done for each s and ε. For these simulations, a free software environment R [31] was used including package Chaos01 developed by Martinoviˇc [32]. The output parameter of the 0–1 test for chaos can acquire only one of the values 0 or 1, which correspond to the regular and chaotic motions, respectively. More details can be found in [29]. The position of the studied system at any moment of time is determined by displacements Fx and Fy , which are used for defining vector φ:  φ( j) =

j

j

Fx (x, y)2 + Fy (x, y)2 .

The output of the 0–1 test for chaos in dependence on s and  is visible in Fig. 2.6. Here, the green mark stands for regular movement, i.e. K is close to zero and the red marks for chaotic one, i.e. K is close to one.

2.5 Conclusions The main goal of this paper was the research of dynamical properties of the Laplaciantype coupled map lattice F (2.3) induced by the Tent family Ts (2.2) on a periodic lattice in dependence on two parameters: the state parameter s of the Tent map and

2 CML-Tent Model Chaotic Behavior with Respect …

25

the coupling constant ε. For this purpose, tools like approximate entropy, maximal Lyapunov exponent, and the 0–1 test for chaos were introduced and applied to numerical simulations performed using a supercomputer located at IT4Innovation National Supercomputing Center, Czech Republic. The necessity of this study follows by open problem stated in [9], where a small region of parameters for which horseshoe phenomena was analytically detected. This research was also motivated by [5, 6] where boundary situation was analytically explored. Due to the complexity of the model (2.3), the remaining part of the parameters region was left for further exploration. The exploration of the model (2.3) was done here using approximate entropy in Fig. 2.4, maximal Lyapunov exponent in Fig. 2.5, and the 0–1 test for chaos in Fig. 2.6 showing systems behavior depending on parameters. It was firstly proved in Theorem 2.3 that for the coupling parameter ε below 1/2 the system (2.3) has only homoclinic trajectories tending to (0, 0). Moreover, in Theorem 2.4 the symmetry around the horizontal line 1/2 was shown. Hence, it is enough to discover dynamics phenomena in the system’s parameter set (s, ε) ∈ (1/2, 1] × (1/2, 1]. The periodic (fixed) structures was pointed out in Theorems 2.1 and 2.2 where it was shown that there are at most four fixed points above 1/2 coupling constant ε. The union of all fixed points throw all parameters values were given in Fig. 2.2. Global attractors with contour plots for specific choice of parameters were also given in Fig. 2.3. As it is visible form Figs. 2.4, 2.5 and 2.6, the results coincide up to few cases that are not decidable. Comparing these results it follows that there is a big set of parameters around (1, 1), or (0, 0) resp., for which chaos was demonstrated. In the other cases the occurrence of some types of intermittencies are assumed, see, e.g., [33, 34] and left for further research.

Fig. 2.7 Periodic points in Is × Is of the model (2.3) for s = 0.9 and ε = 0.9

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

26

M. Lampart and T. Martinoviˇc

To the end, it is easy to check that cardPer 1 (F) = 21 × 2 = 4, cardPer 2 (F) = 2 × 2 = 8, cardPer 3 (F) = 23 × 2 = 16, and cardPer n (F) = 2n × 2 = 2n+1 where Per n (F) denotes the set of all periodic points of period n under F. So, there are two 2-cycles, four 3-cycles, six 4-cycles, twelve 5-cycles etc. They are shown in Fig. 2.7 for s = 0.9 and ε = 0.9; here, black dots are Per 1 in Is × Is , red dots are Per 2 , cyan dots are Per 3 , yellow dots are Per 4 , blue dots are Per 5 . Hence we can naturally ask: 2

Open problem For which s and ε the set of all periodic points of F form dense  subset of Is × Is . Acknowledgements This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science—LQ1602”; by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center—LM2015070”; by the Technology Agency of the Czech Republic (partially by the project TN01000007 “National Centre for Energy” and partially by the project TK02030039 “Energy System for Grids”); by SGC grant No. SP2019/125 “Qualification and quantification tools application to dynamical systems”, VŠB—Technical University of Ostrava, Czech Republic, Grant of SGS No. SP2019/84, VŠB - Technical University of Ostrava, Czech Republic.

References 1. L.A. Bunimovich, E.A. Carlen, J. Differ. Equ. 123, 213 (1995) 2. K. Kaneko, Theory and Applications of Coupled Map Lattices (Wiley, Chichester; New York, 1993) 3. K. Kaneko, Phys. D Nonlinear Phenom. 34, 1 (1989) 4. K. Kaneko, Chaos an interdiscip. J. Nonlinear Sci. 2, 279 (1992) 5. J.L.G. Guirao, M. Lampart, J. Math. Chem. 48, 66 (2010) 6. J.L.G. Guirao, M. Lampart, J. Math. Chem. 48, 159 (2010) 7. F.H. Willeboordse, K. Kaneko, Phys. Rev. Lett. 73, 533 (1994) 8. F.H. Willeboordse, Chaos an interdiscip. J. Nonlinear Sci. 2, 423 (1992) 9. M. Lampart, P. Oprocha, Chaotic sub-dynamics in coupled logistic maps. Phys. D: Nonlinear Phenom. 335, 45–53 (2016) 10. P. Collet, J.P. Eckmann, Iterated Maps on the Interval as Dynamical Systems (Birkhaöuser, Basel; Boston, 1980) 11. M. Lampart, T. Martinovic, Adv. Electr. Electron. Eng. 15, 304 (2017) 12. J. Jost, M.P. Joy, Phys. Rev. E - Stat. Physics, Plasmas, Fluids, Relat. Interdiscip. Top. 65, 16201 (2002) 13. W.-W. Lin, C.-C. Peng, C.-S. Wang, Int. J. Bifurc. Chaos 09, 1635 (1999) 14. B. Fernandez, M. Hang, Ergod. Theory Dyn. Syst. 24, 107 (2004) 15. M. Ding, W. Yang, Phys. Rev. E - Stat. Physics, Plasmas, Fluids, Relat. Interdiscip. Top. 56, 4009 (1997) 16. R. Carvalho, B. Fernandez, R. Vilela Mendes, Phys. Lett. Sect. A Gen. At. Solid State Phys. 285, 327 (2001) 17. G. He, A. Lambert, R. Lima, Phys. D Nonlinear Phenom. 103, 404 (1997) 18. R.L. Viana, C. Grebogi, S.E. de S. Pinto, S.R. Lopes, A.M. Batista, J. Kurths, Phys. D Nonlinear Phenom. 206, 94 (2005) 19. J. Vandermeer, A. Kaufmann, J. Math. Biol. 37, 178 (1998) 20. A.L. Lloyd, J. Theor. Biol. 173, 217 (1995) 21. F. Khellat, A. Ghaderi, N. Vasegh, Chaos. Solitons Fractals 44, 934 (2011)

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22. N. Romero, J. Silva, R. Vivas, J. Math. Anal. Appl. 415, 346 (2014) 23. M. Lampart, T. Martinovic, Chaotic behavior of the CML model with respect to the state and coupling parameters. J. Math. Chem. - first online 24. K.M. Brucks, B. Diamond, M.V. Otero-Espinar, C. Tresser, Dense orbits of critical points for the tent map. Contemp. Math. 117, 57–61 (1991) 25. V.A. Dobrynskii, J. Dyn. Control Syst. 5, 227 (1999) 26. M. Sabeti, S. Katebi, R. Boostani, Artif. Intell. Med. 47, 263 (2009) 27. S.M. Pincus, Proc. Natl. Acad. Sci. 2297–2301 (1991) 28. H. Kantz, Phys. Lett. A 185, 77 (1994) 29. G.A. Gottwald, I. Melbourne, Proc. R. Soc. London A Math. Phys. Eng. Sci. 460, 603 (2004) 30. G.A. Gottwald, I. Melbourne, SIAM, J. Appl. Dyn. Syst. 8, 129 (2009) 31. R Core Team, R: A language and environment for statistical computing. (R Foundation for Statistical Computing, 2018), https://www.R-project.org/ 32. T. Martinovic, Chaos01: 0-1 Test for Chaos. R package version 1.1.1. (2018), https://CRAN. R-project.org/package=Chaos01 33. J.F. Heagy, N. Platt, S.M. Hammel, Phys. Rev. E 49, 1140 (1994) 34. N. Platt, E.A. Spiegel, C. Tresser, Phys. Rev. Lett. 70, 279 (1993)

Chapter 3

Prediction of Echo from Noise Signals by Means of Nonlinear Transform of Signal Spectra Darya E. Apushkinskaya and Evgeny G. Apushkinskiy

Abstract In this note we discuss the algorithm allowing to predict the time of occurrence of echo signals under multi-pulse noise excitation. The technique is verified by the experiments with 59 Co samples.

3.1 Introduction Echo is a phenomenon of emission by the working substance of a time-limited signal occurring after the end of irradiation of external field pulses. The echo signal arises at the moment of restoration of the phased (coherent) state of emitters composing the substance. The echo contains an information about the external pulses affecting the substance, and can be used to convert them in a form suitable to various applications [1]. Modifying the approach introduced in [2], we construct the model of echo excitation, where the frequencies of the input pulses are transformed in a nonlinear manner, and describe the algorithm, which allow us to predict the time of occurrence of echosignals. Among the input pulses irradiating the substance may be signals with noise filling. In this case, to have controlled echo signals at hand may be convenient for storage and reproduction of individual implementations of noise processes as well as for filtering of weak noise signals in the presence of strong noises.

D. E. Apushkinskaya Saarland University, P.O. Box 151150, 66041 Saarbrücken, Germany Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., 117198 Moscow, Russia E. G. Apushkinskiy (B) Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya ul. 29, 195251 St. Petersburg, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_3

29

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D. E. Apushkinskaya and E. G. Apushkinskiy

Fig. 3.1 Schematic view of the input signal S(t)

3.2 Algorithm Let the input signal S(t) be given by a sequence of M (M ≥ 2) real non-overlapping pulses Sm of a finite duration (see Fig. 3.1), i.e., S(t) =

M 

Sm (t − τm ),

t ≥ 0.

(3.1)

m=1

Here τm stands for the beginning of the mth pulse, and for every m we suppose that Sm (t) = 0 for 0 ≤ t < τm . Without loss of generality we assume that τ1 = 0 and 0 < τ2 < · · · < τ M . To apply the Fourier transform to the input signal, we extendthe function S(t) +∞ along negative time-axis by the even reflection. Setting σm (ω) = 0 Sm (t)e−iωt dt and taking into account linearity and time shifting properties of the Fourier transform, we see that the spectrum of the real input signal S(t) can be written as  S(ω) =

M  m=1

 Sm (ω) =

M    σm (ω)e−iωτm + σm∗ (ω)eiωτm dt,

(3.2)

m=1

where σm∗ (ω) is the complex conjugation of σm (ω). (ω), respecFurther, we denote the output signal and its spectrum by U (t) and U (ω) by polynomials tively. Following [2], we approximate U (ω)  U

N  n=0

 n S(ω) , an 

(3.3)

3 Prediction of Echo from Noise Signals by Means of Nonlinear …

31

where an are real constants. The term with n = 0 can be considered as an own thermal noise of our quadrupole, while the term with n = 1 is the standard linear transformation of input signal (e.g., filtration). The subsequent terms in (3.4) with n ≥ 2 are nonlinear frequency transformations, and according to [3, 4] only these summands may be responsible for formation of echo signals. Substituting (3.2) into (3.3) and applying successively the multinomial formula and the inverse Fourier transform we get the following representation for the output signal ∞  M N  

q an n! 1  Sk (ω) k eiωt dω. (3.4) U (t)  2π q1 !q2 ! . . . qm ! k=1 n=0 q −∞

k

 The sum qk in formula (3.4) is taken over all combinations of nonnegative integer indices q1 through q M such that q1 + q2 + · · · + q M = n. Recall that under echo signals we understand the supplementary (new) signals occurring at time axis in addition to the base (input) ones. It is evident that for echoes occurring after irradiation of M pulses, M ≥ 2, the following conditions are fulfilled: The maximum value of echo signal is realized when t > τ M + Δ M , where Δ M is the duration of Mth pulse;

(3.5)

All pulses S1 , . . . , S M are participated in formation of echo.

(3.6)

We claim that there are no echo signals if the parameters M and N in (3.4) satisfy either the inequality N < M or the equalities M = N = 2. Indeed, the first statement follows immediately from the condition (3.6). To verify the second statement, we set M = N = 2 in formula (3.4) and arrive at 1 U (t)  2π

∞

    2  a0 + a1  S1 +  S1 + 2 S2 + a2  S1 S22 eiωt dω. S2 + 

(3.7)

−∞

Taking into account (3.6), we see that the term 2 S1  S2 in (3.7) can be the only candidate for generating an echo signal. In view of the second equality in (3.2) we have

 S2 (ω)eiωt = (σ1 + σ1∗ ) σ2 eiω(t−τ2 ) + σ2∗ eiω(t+τ2 ) . S1 (ω)

(3.8)

If the terms in the right-hand side of (3.8) generate an echo signal then its maximum value is achieved at t = τ2 , but the latter contradicts (3.5). Now, we show in several examples how to detect an echo signal from representation (3.4).

32

D. E. Apushkinskaya and E. G. Apushkinskiy

For M = 2 and N = 3, the output signal is expressed by 1 U (t)  r.h.s. (7) + 2π

∞

 3  S1 + 3 a3  S12  S1 S23 eiωt dω. S2 + 3 S22 + 

−∞

Due to (3.6) and (3.2) we obtain that the echoes can be provided only by the terms  2     S1  S1 S2 +  S22 eiωt = (σ1 + σ1∗ )2 σ2 eiω(t−τ2 ) + σ2∗ eiω(t+τ2 )   + (σ1 + σ1∗ ) σ22 eiω(t−2τ2 ) + 2σ2 σ2∗ eiωt + (σ2∗ )2 eiω(t+2τ2 ) .

(3.9)

The term (σ1 + σ1∗ )σ22 eiω(t−2τ2 ) in (3.9) generates the echo signal with maximal value occurring at t = 2τ2 , which satisfies (3.5). For other summanden in (3.9) we get a contradiction with restriciton (3.5). Consider now the case of three input pulses, i.e. M = 3. Setting N = 3 one can S2  S3 from the right-hand side of (3.4) satisfies the condition see that only the term  S1 (3.6). Again, in view of (3.2), we get   S1 S2  S3 eiωt = (σ1 + σ1∗ ) σ2 σ3 eiω(t−τ2 −τ3 ) + σ2 σ3∗ eiω(t−τ2 +τ3 )  + σ2∗ σ3 eiω(t+τ2 −τ3 ) + σ2∗ σ3∗ eiω(t+τ2 +τ3 ) , and conclude that the only term (σ1 + σ1∗ )σ2 σ3 eiω(t−τ2 −τ3 ) generates the echo signal with maximal value occuring at t = τ2 + τ3 , which satisfies (3.5). Arguing in the same manner one can predict the occurrence the echo signals for an arbitrary number of input pulses. For the readers convenience we provide in Table 3.1 the results of our calculation for M = 2 and 3, and for N = 3, 4, and 5, respectively. To sum up the above results, we extract from the right-hand side of (3.4) the terms providing the echo signals E M (t) after irradiation of M pulses, M ≥ 2, and end up with the following approximations: 1 E2 (t)  2π E M (t) 

∞  N



p

p S1 (ω) 1  S2 (ω) 2 eiωt dω, bn 

−∞ n=3 ∞ N

1 2π



−∞ n=M

bn

M 

p  Sk (ω) k eiωt dω,

M = 2,

M ≥ 3,

(3.10)

(3.11)

k=1

where pk are integer indices ( pk ≥ 1), such that p1 + · · · + p M = n, while bn are real constants depending on pk ! and an .

3 Prediction of Echo from Noise Signals by Means of Nonlinear …

33

Table 3.1 The times of maximums and spectra of echo signals occurring after irradiation of two and three pulses M N 3 4 5 t Spectrum∗ t Spectrum∗ t Spectrum∗ 2

3

2τ2

τ2 + τ3

(σ1 + σ1∗ )σ22

(σ1 + σ1∗ )σ2 σ3

2τ2

(σ1 + σ1∗ )2 σ22

2τ2

3τ2

(σ1 + σ1∗ )σ23

3τ2

τ2 + τ3

(σ1 + σ1∗ )2 σ2 σ3

τ2 + 2τ3

(σ1 + σ1∗ )σ2 σ32 (σ1 + σ1∗ )σ2∗ σ32 (σ1 + σ1∗ )σ22 σ3

2τ3 − τ2 2τ2 + τ3

∗

(σ1 + σ1∗ )3 σ22 + 4(σ1 + σ1∗ )σ23 σ2∗ (σ1 + σ1∗ )2 σ23

τ2 + 2τ3

(σ1 + σ1∗ )σ24 (σ1 + σ1∗ )σ2 σ3 [(σ1 + σ1∗ )2 + 3(|σ2 |2 + |σ3 |2 )] (σ1 + σ1∗ )2 σ2 σ32

2τ3 − τ2

(σ1 + σ1∗ )2 σ2∗ σ32

2τ2 + τ3

(σ1 + σ1∗ )2 σ22 σ3

3τ2 + τ3 3τ2 − τ3 ∗∗ 3τ3 + τ2 3τ3 − τ2 2τ2 + 2τ3 2τ3 − 2τ2 ∗∗∗ 2τ3 2τ2 ∗∗∗∗

(σ1 + σ1∗ )σ23 σ3 (σ1 + σ1∗ )σ23 σ3∗ (σ1 + σ1∗ )σ2 σ33 (σ1 + σ1∗ )σ2∗ σ33 (σ1 + σ1∗ )σ22 σ32 (σ1 + σ1∗ )(σ2∗ σ3 )2

4τ2 τ2 + τ3

(σ1 + σ1∗ )|σ2 |2 σ32 (σ1 + σ1∗ )σ22 |σ3 |2

The spectra are given without coefficients an and constants from the multinomial formula; ∗∗ if τ3 < 23 τ2 ; ∗∗∗ if τ3 < 2τ2 ; ∗∗∗∗ if τ3 < 2τ2 . With increasing of N , the spectra for the equal times should be added, e.g. for M = 2 and N = 4 the spectrum of the echo occurring at t = 2τ2 is the sum of the corresponding values in the 3rd and 5th columns, i.e., (σ1 + σ1∗ )σ22 [1 + σ1 + σ1∗ ]

3.3 Experimental Results We observed a nuclear spin echo in thin cobalt films (isotope 59 Co) after applying a series of two or three pulses. One pulse in every series was so short (0.2 µs in our case) that its spectrum overlapped the film absorption band. This short pulse had the basic harmonic frequency coinciding with the central absorption frequency for cobalt film equal to 216 MHz. The rest pulses of the series had noise carrier with duration of several microseconds. Figures 3.2 and 3.3 show the examples of two-pulse echoes. In both cases the maximal values of echo signals occur at time t = 2τ2 , i.e., at t = 12 µs and t = 8 µs,

34

D. E. Apushkinskaya and E. G. Apushkinskiy

Fig. 3.2 Echo signal from a train of two RF pulses. Delay between pulses τ2 = 6 µs; 1st pulse with noise signal (Δ1 = 1 µs), 2nd pulse with harmonic signal (Δ2 = 0.2 µs)

respectively. Figure 3.4 illustrates the examples of two- and three-pulse echo signals. Here, the two-pulse echo occurs at t = 2τ2 = 6 µs, while the three-pulse echo is detected at t = τ2 + τ3 = 10 µs. Thus, the experimental and theoretical results are in good agreement (cf. Table 3.1).

3.4 Conclusion Comparison of theoretical and experimental results shows that the proposed algorithm provide the correct results for different types of exciting pulses, in particular, for noise pulses. Such versatility provides the relevance of this algorithm for simulations of echo processes without reference to equations describing the corresponding models. It should be emphasized that analysis of nonlinear stochastic differential equations modelling the echo processes upon application of noise external actions is extremely difficult. This fact can strongly complicate the prediction of echoes appearance. Our method is free of this complication.

3 Prediction of Echo from Noise Signals by Means of Nonlinear …

35

Fig. 3.3 Echo signal from a train of two RF pulses. Delay between pulses τ2 = 4 µs; 1st pulse with harmonic signal (Δ1 = 0.2 µs), 2nd pulse with noise signal (Δ2 = 1 µs)

Moreover, if the working substance is excited by weak signals lying below the noise level, one can arrange the input signals on the time axis such that the echo will be strongly amplified and, consequently, will exceed the noise. The inverse problem—to establish the presence or absence of excitation pulses by the time of occurrence of echo signals—can be solved as well.

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D. E. Apushkinskaya and E. G. Apushkinskiy

Fig. 3.4 Echo signal from a train of three RF pulses. Delay between pulses τ2 = 3 µs and τ3 = 7 µs; 1st and 3rd pulses with noise signal (Δ1 = Δ3 = 2 µs), 2nd pulse with harmonic signal (Δ2 = 0.2 µs)

Acknowledgements D.A. was partly supported by the “RUDN University Program 5-100” and the RFBR grant 18-01-00472.

References 1. I.V. Pleshakov, P.S. Popov, V.I. Dudkin, YuI Kuz’min, Spin echo processor in functional electronic devices: control of responses in processing of multipulse trains. J. Commun. Technol. Electron. 62(6), 583–587 (2017) 2. E.G. Apushkinskiy, Nonlinear transformation of signal spectra. Nauch. Techn. Vedomosti SPbSPU 3(153), 182–190 (2012) (Russian) 3. A. Korpel, M. Chatterjee, Nonlinear echoes, phase conjugation, time reversal, and electronic holography. Proc. IEEE 69(12), 1539–1556 (1981) 4. I.S. Ryzak, General patterns of causal nonlinear echo formation and their application to multifunctional signal processing. Radiotech. I Electron. 45(1), 5–38 (2000) (Russian)

Chapter 4

Fractal Functions and the Dragon’s Mountain: A Functional Equations Perspective Cristina Serpa

Abstract We use an iterative method to construct solutions of iterated functions systems. This suggests a characterization of fractal functions via an admissibility alphabet. In this context, we define a new fractal object called dragon’s mountain, a real valued function with a complex domain designated by dragon’s set. The fractal structure is induced on solution sets by this type of structure, using the tools of symbolic dynamics. Mathematics Subject Classification: Primary 30D05 · 37F99; Secondary 30E10 · 30L99 · 30E99 · 41A05 · 58C05

4.1 Introduction Fractal data modeling represents a radical shift of perspective from classical data analysis. Traditional data models are smooth, fitting all type of phenomena with nice analytic functions. Typically, data is interpolated using, polynomial, trigonometric, spline, or Hermite methods. Recently, in order to better fit chaotic phenomena, without smooth behavior, scientists introduced fractal interpolation methods to estimate data. The first and simplest model (a system of functional equations based on affine functions with constant coefficients) is due to Barnsley [1]. This method has been quickly and widely implemented in numerous fields of applied science, namely Physics, Engineering, Hydrology, Climatology, Seismology, Medicine, Biology and Economics. The standard to this method is the case-by case study of interpolations, simulating numerically estimated fractals. Recently explicit and constructive solutions for fractal interpolation functions are given, not only for the affine case [12], but also for generalized nonlinear cases [11]. These papers provide a pointwise exact calculation, a very significant progress with respect to the standard numeriC. Serpa (B) Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Instituto Superior de Engenharia de Lisboa, Lisbon, Portugal e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_4

37

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C. Serpa

cal approach. Paper [11] constructs a theoretical framework for general systems of functional equations whose solutions are fractal interpolation functions, thus establishing the mathematical formalization of necessary and sufficient conditions for existence of solution and conditions on continuity (in the context of metric spaces and families of contractive functions). The concept of compatibility conditions (for a system to be well-defined) is formally introduced, which were previously and implicitly used by authors in a case-by-case manner, without any general and theoretical formalization. The constructions made in [11] for fractal interpolation functions are obtained as solutions of a type of system of iterative functional equations. Fractal functions are defined originally by Barnsley [1] and called fractal interpolation functions. These functions are solutions of iterative functional equations and have their roots in the theory of iterated function systems (see, e.g., [6]). The term “interpolation” aims to be a valid instrument to approximate real non-smooth data with a mathematical formulation. The use of fractal functions in applications is widening its spectrum of scientific fields. The beauty of the fractals objects has gained visibility from Mandelbrot (see [8, 9]). Let X and Y be non-empty sets and p ≥ 2 be an integer. Consider a system of functional equations ϕ ( f k (x)) = Fk (x, ϕ (x)) , x ∈ X, k = 0, 1, . . . p − 1,

(4.1)

where f k : X → W ⊂ X , Fk : X × Y → Y are given functions, and ϕ : X → Y is the unknown function. This system is a generalization of the Barnsley [1] fractal interpolation functions setting. There are necessary conditions for the existence of solutions for the system (4.1) called compatibility conditions. These conditions were introduced by Serpa-Buescu in [11, 12] and are stated in the following (for f k injective). Proposition 1 If ϕ : X → Y is a solution of (4.1), then it must satisfy ∀x1 , x2 ∈ X, f k (x1 ) = f j (x2 ) ⇒ Fk (x1 , ϕ (x1 )) = F j (x2 , ϕ (x2 )) ,

(4.2)

for k, j = 0, 1, . . . , p − 1. Let A :=

p−1  

 x1 ∈ X : ∃ j = 0, . . . , p − 1, j = i, ∃ x2 ∈ X, f i (x1 ) = f j (x2 ) .

i=0

The elements of A are called the contact points of system (4.1). Definition 2 Consider a system of equations (4.1) where for all x ∈ A, ϕx ≡ ϕ (x) has been previously determined (by partially solving the system or by initial conditions). We say that     ∀x1 , x2 ∈ A, f k (x1 ) = f j (x2 ) ⇒ Fk x1 , ϕx1 = F j x2 , ϕx2 ,

(4.3)

4 Fractal Functions and the Dragon’s Mountain: A Functional Equations Perspective

39

are the compatibility conditions of system (4.1). Compatibility conditions are necessary but not sufficient for the existence of solutions. Let (X, d1 ) be a bounded metric space and (Y, d2 ) be a complete metric space. Definition 3 Let f : X → Y . We say f is a contraction map if there is some nonnegative real number 0 < λ < 1 such that for all x1 and x2 in X , d2 ( f (x1 ) , f (x2 )) ≤ λd1 (x1 , x2 ). In this case λ is called the contraction factor and f is called a λcontraction. The general existence and uniqueness result [11] is given as following. Theorem 4 Let f k : X → X be a family of injective functions such that (i) ∀i = j, f k (X ) ∩ f j (X ) = ∅, p−1 (ii) ∪i=0 f k (X ) = X and Fk : X × Y → Y is, with respect to the second coordinate, a λk -contraction, for all k = 0, 1, . . . , p − 1. Then there exists a unique bounded solution ϕ : X → Y of system (4.1). The representation of the elements of the set X as sequences of symbols plays a crucial role in obtaining the solutions of systems (4.1). The natural numbers may be represented (see [5]) in any integer base p > 1 using digits 0, 1, . . . , p − 1 by k  ξn p n , (4.4) x= n=1

where k ∈ N, and the real numbers x ∈ [0, 1] by x=

∞  ξn n=1

pn

.

(4.5)

The representation of all real numbers is a combination between (4.4) and (4.5). The usual bases are the binary ( p = 2) and decimal ( p = 10). The case of complex numbers has some peculiarities. A Gaussian integer [3] is a complex number z = x + i y, where x and y are real integers. Kátai and Szabó [7] proved that the only Gaussian integers that can be used as a base for all the complex numbers, using natural numbers as digits, are − p + i and − p − i. The digits for these representations are 0, 1, 2, . . . , p 2 . Analogously as for real numbers, complex numbers may be written using the base − p + i (see [4, 13]), such that z=

∞ 

ξn , p + i)n (− n=−k

(4.6)

40

C. Serpa

Fig. 4.1 Dragon’s set—points up to five digits

  where ξn ∈ 0, 1, 2, . . . , p 2 . Such a representation is called radix-representation. Let D ⊂ C be the dragon’s set, defined as the set of complex numbers whose integer part, in the base −1 + i, is zero. This set is usually called dragon curve (see e. g. [2]). The elements of D with one digit are (0.0)−1+i = 0 and (0.1)−1+i = (−1 + i)−1 = − 21 − 21 i. The points of D up to five digits are depicted in Fig. 4.1 and with a precision of eight digits in Fig. 4.2. The boundary of D is itself a fractal (see [4]), with fractal dimension equal to 1.523627085 (see [2]). We define an object, the dragon’s mountain, that is the graph of a special case of system (4.1) with domain X = D. Definition 5 The dragon’s mountain is the graphic of a real valued function ϕ : D ⊂ C → R, with complex domain D, defined as the solution of system 

z+k ϕ −1 + i

= sk ϕ (z) + rk |z| + tk , z ∈ D, k ∈ {0, 1} ,

(4.7)

where rk : D → R, sk , tk ∈ R and 0 < |sk | < 1. The construction of a dragon’s mountain is a generalization of fractal interpolation functions, extending the domain of ϕ to a subset of C—the dragon’s set. Proposition 6 Let p = −1 + i and z=

∞  ξn n=1

pn

.

(4.8)

4 Fractal Functions and the Dragon’s Mountain: A Functional Equations Perspective

41

Fig. 4.2 Dragon’s set—points up to eight digits

If a continuous solution ϕ of (4.7) exists, then ϕ is given, in terms of the base p expansion of z, by ϕ (z) =

n−1 ∞   n=1

m=1

∞



 ξ k+n

rξn

+ tξn .

pk



sξm

(4.9)

k=1

Proof This is a corollary of a constructive formula for the solutions of systems (4.1) given in [11].  The existence of solution of system (4.7) requires that the compatibility conditions be satisfied. In this case, these conditions are ∀z 0 , z 1 ∈ D, z 0 = z 1 + 1 ⇒ s0 ϕ (z 0 ) + r0 |z 0 | + t0 = s1 ϕ (z 1 ) + r1 |z 1 | + t1 . From this conditions follows the determination of parameters s0 , r0 , t0 , s1 , r1 , t1 . However, the effective computation of compatible parameters is an open problem. In the situation of unknown compatible parameters, it is possible to construct a discrete dragon’s mountain, which satisfies system (4.7) up to a digit level. For that purpose, we choose a set of parameters s0 , r0 , t0 , s1 , r1 , t1 and a digit level and apply a discrete formula. Proposition 7 Let p = −1 + i and n ∈ N. Then, a discrete solution ϕ of (4.7) up to digit n exists and is given, in terms of the base p expansion of z, by

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C. Serpa

  ϕ (0.ξ1 ξ2 · · · ξn ) p =

n  m=1

 

t0 + sξm tξk + sξm rξn−k+1 (0.ξn−k+2 · · · ξn ) p , (4.10) 1 − s0 k=1 m=1 k=2 m=1 n k−1

sξm

n n−k

where ξk ∈ {0, 1}, k ∈ {1, 2, . . . , n − 1} and ξn = 1. The graph of ϕ is called discrete dragon’s mountain. Proof First we obtain the image of point 0. From (4.7), with z = 0 and k = 0, we obtain t0 ϕ (0) = . 1 − s0 The proof for the general case proceeds by induction. For n = 1, given ϕ (0) and (4.7), with z = 0 and k = 1, we have 

ϕ (0.1) p





0+1 =ϕ −1 + i

= s1

t0 + t1 . 1 − s0

For the induction step, let n ∈ N. Suppose that (4.10) is valid for (0.ξ1 ξ2 · · · ξn ) p . Since (0.ξ2 ξ3 · · · ξn+1 ) p + ξ1 , (0.ξ1 ξ2 · · · ξn ξn+1 ) p = p Equation (4.7) becomes  ϕ

(0.ξ2 ξ3 · · · ξn+1 ) p + ξ1 p



  = sξ1 ϕ (0.ξ2 ξ3 · · · ξn+1 ) p + rξ1 (0.ξ2 ξ3 · · · ξn+1 ) p + tξ1 .

Through straightforward computations, we obtain the result for n + 1. Therefore formula (4.10) is valid for all n ∈ N.  We now provide an example. Consider system (4.7), where n = 8, s0 = s1 = 0.5, r0 = −0.75, r1 = −0.25, t0 = −0.1 and t1 = −0.5. The corresponding discrete dragon’s mountain is obtained by formula

=−

1 15 · 2n−1

  ϕ (0.ξ1 ξ2 · · · ξn ) p

k−1

n−k n n  1  1

+ tξk + rξn−k+1 (0.ξn−k+2 · · · ξn ) p 2 2 k=1 m=1 k=2 m=1

and is a 3D object as illustrated in Fig. 4.3. We observe that there is a song with the name Dragon’s Mountain by the italian composer Psychevision (see [10]). The background image of the associated video is a mountain that resembles the graph of the function depicted in Fig. 4.3.

4 Fractal Functions and the Dragon’s Mountain: A Functional Equations Perspective

43

Fig. 4.3 Discret dragon’s mountain—points up to eight digits

Acknowledgements The author acknowledges partial support by National Funding from FCT— Fundação para a Ciência e a Tecnologia, under the project: UID/MAT/04561/2019.

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References 1. M. Barnsley, Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986) 2. A. Chang, T. Zhang, The fractal geometry of the boundary of dragon curves. J. Recreat. Math., 30(1), 9–22 (1999-2000) 3. C.F. Gauss, Theoria residuorum biquadraticorum, Commentatio secunda. Comm. Soc. Reg. Sci. Göttingen 7, 1–34 (1832) 4. W.J. Gilbert, Fractal geometry derived from complex bases. Math. Intell. 4, 78–86 (1982) 5. G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, 5th edn. (Oxford Science Publications, 1979) 6. J. Hutchinson, Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981) 7. I. Kátai, J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37, 255–260 (1975) 8. B.B. Mandelbrot, Fractals: Form (Chance and dimension, Freeman, San Francisco, CA, 1977) 9. B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, CA, 1982) 10. Psychevision, https://www.youtube.com/watch?v=ef1IJFQW4cI, Accessed in 26 April 2019 11. C. Serpa, Buescu, constructive solutions for systems of iterative functional equations. Constr. Approx. 45, 273–299 (2017) 12. C. Serpa, J. Buescu, Explicitly defined fractal interpolation functions with variable parameters. Chaos Solitons Fractals 75, 76–83 (2015) 13. A. Vince, Rep-tiling euclidean space. Aequationes Math. 50, 191–213 (1995)

Chapter 5

Movement Characteristics of a Model with Circular Equilibrium Marek Lampart and Judita Nagyová

Abstract In this paper, the dynamical behavior of the Gotthans-Petržela third-order autonomous model is researched. For this purpose, the 0-1 test for chaos and fast approximate entropy is newly applied. Using these tools, the dynamic is quantified and qualified. Depending on the system’s parameters, it is shown that irregular (chaotic) and regular (periodic) movement character appears.

5.1 Introduction The study of dynamical systems has attracted researchers in various fields for more than a century. It has been discovered that chaotic behavior is natural to not only complicated nonlinear vector fields but to dynamical systems in general. Irregular behavior can also be observed in algebraically simple models like one-dimensional discrete systems [2]. With major improvement in the field of parallel computation, it is possible to search for chaos in nonlinear dynamical systems by tools coming from the theory of dynamical system under a choice of parameters and initial conditions. As chaos is often associated with singular saddle-type fixed points [4], the research on systems with equilibrium with specific properties has drawn interest. In this way, it is possible to come across dynamical systems with very specific properties. As the first step of model dynamics search, the equilibria are studied. For example, in [10] or [22], a dynamical system with no equilibrium was presented. On the other hand, there exist a number of dynamical systems with infinite equilibrium, which can form different shapes, such as a line [23], square or rectangle [5, 24], a curve M. Lampart · J. Nagyová (B) IT4Innovations, VŠB—Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, Czech Republic e-mail: [email protected] M. Lampart e-mail: [email protected] Department of Applied Mathematics, VŠB—Technical University of Ostrava, 17. listopadu 2172/15, 708 33 Ostrava, Czech Republic © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_5

45

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[26, 30], or a circle [4, 23]. The more unusual ones involve a shape of a cloud [35], an axe [36], a boomerang [20], a clover [21], or a pear [28]. These dynamical systems are motivated by real problems arising in engineering, chemistry, economy, biology, etc. The model of this paper is motivated by an electrical circuit. This work focuses on the characterization of dynamical properties of a model with circular equilibrium [4], see also [5, 12, 24, 25]. The characteristic is done using mainly the 0-1 test for chaos and fast approximate entropy. These tools are applied to the huge simulation data that was performed on the Salomon supercomputer at IT4Innovations National Supercomputing Center located in Ostrava, Czech Republic. The paper is organized as follows: in Sect. 5.2 the researched model is introduced, in Sect. 5.3 main results are reached using the 0-1 test for chaos, fast approximate entropy, and the Fourier spectra analysis, and finally in Sect. 5.4 main results are summarized and simulation details are given.

5.2 The Model The investigated model was introduced by Gotthans and Petržela in [4] as a continuous third-order autonomous deterministic dynamical system. This model is motivated by a real occurring problem, namely the lumped electronic circuit, and is introduced as a system of three differential equations in its dimensionless form: dx = az, dt dy = z f 1 (x, y, z), dt dz = x 2 + y 2 − r 2 + z f 2 (x, y, z), dt

(5.1)

where r is the radius of circular equilibrium and a a free parameter. The nonlinear functions f 1 and f 2 are chosen as f 1 (x, y, z) = bx + cz 2 f 2 (x, y, z) = d x, where b, c, and d are free parameters. It is easy to see that the set of equilibria forms a circle on the plane z = 0, and a local behavior along this equilibrium circle was discussed using the so-called eigenvalues technique in [4]. It was also shown in [4] that the model’s trajectory character admits chaotic movement as well as periodic one. This was done using a concept of the largest Lyapunov exponent.

5 Movement Characteristics of a Model with Circular Equilibrium

47

Contrary to that concept, alternative techniques were applied in this paper. Namely, the 0-1 test for chaos and fast approximate entropy underlined by phase diagrams, the Fourier spectra analysis, and bifurcations diagrams. Those two mentioned dynamical tools for qualification and quantification of movement character were utilized due to their effectivity and robustness compared to the maximal Lyapunov exponent.

5.3 Main Results The major outcomes of this paper were reached by simulations for free parameters a and d in a range of a ∈ [−0.5, −0.1] and d ∈ [−0.5, −0.1] with 0.01 step. More precisely, simulations were performed using the Runge–Kutta fifth-order integration method in Matlab [32] with final time 106 and time step 0.1. To avoid the system’s distortions, the start of the system simulations were skipped, more precisely, the first 8 × 106 records were overleaped, i.e. the last 20% of each simulation were used for further investigation. For the computations of the 0-1 test for chaos and fast approximate entropy, the first 8 × 104 records were skipped and the data were downsampled by 1000. The system is assumed to be in the rest position in the beginning. That is, initial conditions equal (x0 , y0 , z 0 ) = (0, 0, 0). From the main results, a phase diagram, amplitude-frequency spectrum (FFT), and Poincaré section for a relevant choice of a given free parameter were done. Moreover, to underline dynamics behavior, the bifurcation diagrams with a suitable magnification with respect to the free parameter a was built in. Consequently, for a range of parameters a ∈ [−0.5, −0.1] and d ∈ [−0.5, −0.1] the 0-1 test for chaos and fast approximate entropy were computed. The 0-1 test for chaos splits the region of parameters for which regular (periodic or quasi-periodic) and irregular (chaotic) movements appear. On the other hand, the output of the fast approximate entropy detects the complexity of the system depending on the parameters reaching a decrease in prediction while the value of fast approximate entropy increases.

5.3.1 Phase Diagrams, the Fourier Spectra, and Bifurcation Diagrams The movement character of the model (5.1) can be periodic as well as chaotic. For example, in Fig. 5.1 phase diagram together with Poincaré section for a = −0.2 and d = −0.5 nontrivial loop is shown. This corresponds to the periodic case. On the other hand, chaotic movement is given in Figs. 5.2 and 5.3.

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Fig. 5.1 Phase diagram with Poincaré section and Fast Fourier transform of variable x for a = −0.2 and d = −0.5

Fig. 5.2 Phase diagram with Poincaré section and Fast Fourier transform of variable x for a = −0.1 and d = −0.1

Fig. 5.3 Phase diagram with Poincaré section and Fast Fourier transform of variable x for a = −0.2 and d = −0.35

The Fourier spectra and phase portraits were computed for a = −0.2, d = −0.5, a = −0.1, d = −0.1, and a = −0.2, d = −0.35 shown in Figs. 5.1, 5.2, and 5.3 respectively. Regular behavior is observable for the first case and chaos in the last two cases. In the case of regular movement, in Fig. 5.1, the Fourier spectra is formed by a number of harmonic frequencies, hence the frequency of the periodic trajectory is computable. Periodic motions of the trajectory are also visible in Poincaré sections. In the case of irregular movement, Figs. 5.2 and 5.3, the Fourier spectra is formed by a number of harmonic components having the basic, super-harmonic, sub-harmonic, and combination frequencies on which further motions with frequencies forming the sided bands of the dominant frequencies are superposed. Their mutual ratio indicates the irregularity of the motion. The character of this motion’s case is underlined by the Poincaré section. Next, bifurcation diagrams of the model (5.1) were done with respect to the free parameter a for variable x, y, and z in Figs. 5.4, 5.5, and 5.6, respectively. The other constants are chosen as b = −3, c = −2.2, d = −0.1 and r = 1. The left part of mentioned figures corresponds to the range a ∈ (−0.5, −0.1) and

5 Movement Characteristics of a Model with Circular Equilibrium

49

Fig. 5.4 Bifurcation diagram of x with respect to a: left a ∈ (−0.5, −0.1), right a ∈ (−0.15, −0.1)

Fig. 5.5 Bifurcation diagram of y with respect to a: left a ∈ (−0.5, −0.1), right a ∈ (−0.15, −0.1)

Fig. 5.6 Bifurcation diagram of z with respect to a: left a ∈ (−0.5, −0.1), right a ∈ (−0.15, −0.1)

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right to a ∈ (−0.15, −0.1); the right parts are magnifications. In these details the “period doubling” effect with “windows” is visible. For this free parameter, periodic movement is observable if a ∈ (−0.5, −0.14) and chaotic cases appear for some a ∈ (−0.14, −0.1).

5.3.2 Fast Approximate Entropy Fast approximate entropy (FastApEn), proposed by Tomˇcala in [33], is a modification of the approximate entropy (ApEn) algorithm, introduced in [27], leading to its acceleration. It is determined for a given time-series data φ(1), φ(2), . . . φ(N ), positive integer m, and positive real number r by the formula: N  Nm+1     m  |ci,m | |ci,m+1 | 1  1 Fast ApEn(v, m, r ) = log log − Nm i=1 Nm Nm+1 i=1 Nm+1   Nm+1 = μ (log (|cm |)) − μ (log (|cm+1 |)) + log Nm where v(i) = [φ(i), φ(i + 1), . . . , φ(i + m − 1)] is a sequence of vectors in Rm , v(1), v(2), …, v(M − m + 1) for each 1 ≤ i ≤ N − m + 1,   

/ c j,m , j < i , yi = [vi , vi+1 , . . . , vi+m−1 ], ci,m = ξ | ||yi − yξ || ≤ r ∧ ξ ∈ cm are value classes of subsequences of length m and Nm is the number of these value classes, and μ stands for mean. The position of the studied system (5.1) at any moment of time is determined by displacements x, y, and z, which are used for defining vector φ: φ( j) =



x( j)2 + y( j)2 + z( j)2 .

For these simulations, a free software environment R [31] was used including the FastApEn package developed by Tomˇcala [34]. The output of fast approximate entropy depending on a and d is visible in Fig. 5.7. Here, the correlation between neighboring parameter tests detects an increase (or decrease) of the system’s complexity.

5.3.3 The 0-1 Test for Chaos The 0-1 test for chaos, invented by Gottwald and Melbourne [6], is one of the methods for distinguishing between regular and chaotic dynamics of a deterministic system.

5 Movement Characteristics of a Model with Circular Equilibrium

51

Fig. 5.7 Output of fast approximate entropy

In contrast to the other approaches, the nature of the system is irrelevant, thus the test can be applied directly onto experimental data, ordinary differential equations, or partial differential equations. The results return values close to 0 or 1, with 0 corresponding to regular dynamics and 1 to chaotic dynamics. With its easy implementation, evaluation, and wide range of application, using this tool for detecting chaos is becoming more popular in different fields. It was used on data of traffic speed [18], on solar irradiance data [11], or a chaotic bit generator [19], a jumping ball [13], vibrational energy harvesters [9, 16], and oscillators [1, 3, 15, 29]. The 0-1 test for chaos was also applied for non-autonomous continuous dynamical systems, see, e.g. [14]. The 0-1 test for chaos can be computed by the following algorithm [7]. Given the observation φ( j) for j = 1, 2, . . . , N and a suitable choice of c ∈ (0, 2π ), the following translation variables are computed: pc (n) =

n 

φ( j) cos( jc),

j=1

qc (n) =

n 

φ( j) sin( jc)

j=1

for n = 1, 2, . . . , N . The idea for the 0-1 test, first described in [6], is that the boundedness or unboundedness of the trajectory {( p j , q j ) j∈[1,N ] } can be studied through the asymptotic growth rate of its time-averaged mean square displacement, which is defined as M(n) = lim

N →∞

N 1  d( j, n)2 N j=1

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where d( j, n) =

( p j+n − p j )2 + (q j+n − q j )2

is the time lapse of the duration n (n  N ) starting from the position at time j. As it is shown in [7, 8], it is important to use values of n small enough compared to N , noted n cut , (n ≤ n cut ). A subset of time lags n cut ∈ [1, N /10] is advised for the computation of each K c . M(n) is the average of the displacements which can be different depending on the part of the trajectory. It is evaluated over all time lags. If the time lag is fixed, the mean square displacement describes only the local behavior of the trajectory. In order to determine the overall unboundedness, larger values of n must be studied. For bounded trajectories and regular dynamics, M(n) is a bounded function in time, whereas unbounded trajectories, meaning chaotic dynamics, are described by M(n) growing linearly with time. Thus the asymptotic growth rate of the MSD must be calculated, which correlates with the unboundedness of the trajectory. As proposed in [7], the modified MSD is calculated as D(n) = M(n) − E(φ)2

1 − cos(nc) 1 − cos c

The output of the 0-1 test for chaos is computed by the correlation method as K c = corr(ξ, Δ) ∈ [−1, 1] for the vectors ξ = (1, 2, . . . , n cut ) and Δ = (Dc (1), Dc (2), . . . , Dc (n cut )). The final result of the test is K = median(K c ). The position of the studied system (5.1) at any moment of time is determined by displacements x, y, and z, which are used for defining vector φ: φ( j) =



x( j)2 + y( j)2 + z( j)2 .

For these simulations, a free software environment R [31] was used including the Chaos01 package developed by Martinoviˇc [17]. The output of the 0-1 test for chaos depending on a and d is visible in Fig. 5.8.

5 Movement Characteristics of a Model with Circular Equilibrium

53

Fig. 5.8 Output of the 0-1 test for chaos

0

5.4 Conclusions In this paper, the Gotthans-Petržela model (5.1), introduced in [4], was constructed and its dynamics depending on two free parameters a and d deeply researched. The equations of motion were simulated in Matlab [32] using the Runge–Kutta fifth-order integration method for each free parameter setting. The dynamics of the model (5.1) is showing regular as well as irregular patterns. This characteristics were observed using phase portraits, the Fourier spectra (see Figs. 5.1, 5.2, and 5.3), and bifurcation diagrams (see Figs. 5.4, 5.5, and 5.6). The fast approximate entropy and the 0-1 test for chaos were utilized for qualification and quantification of movement character. In detail, in Fig. 5.7 there is the output of the fast approximate entropy showing the system’s complexity, i.e. it is possible to compare neighboring simulations in terms of dynamics. While in Fig. 5.8 the output splits the region of free parameters a and d into two parts. Blue stands for systems regular movement and red for chaotic one. For the evaluation of simulations a free software environment R [31] was used including the Chaos01 [17] and FastApEn [34] packages. Acknowledgements This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science—LQ1602”; by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center—LM2015070”; by the Technology Agency of the Czech Republic (by the projects TN01000007 “National Centre for Energy”, TK02030039 “Energy System for Grids”, and TJ02000157 “Optimization of the electrical distribution system operating parameters using artificial intelligence”); by SGC grant No. SP2019/125 “Qualification and quantification tools application to dynamical systems”, VŠB—Technical University of Ostrava, Czech Republic, Grant of SGS No. SP2019/84, VŠB—Technical University of Ostrava, Czech Republic.

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24. V.-T. Pham, S. Jafari, X. Wang, J. Ma, A chaotic system with different shapes of equilibria. Int. J. Bifurc. Chaos Appl. Sci. Eng. 26 (2016) 25. V.-T. Pham, C. Volos, S. Jafari, S. Vaidyanathan, T. Kapitaniak, X. Wang, A chaotic system with different families of hidden attractors. Int. J. Bifurc. Chaos Appl. Sci. Eng. 26 (2016) 26. V.-T. Pham, C. Volos, T. Kapitaniak, S. Jafari, X. Wang, Dynamics and circuit of a chaotic system with a curve of equilibrium points. Int. J. Electron. 1–13 (2017) 27. S.M. Pincus, Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. 88, 2297–2301 (1991) 28. A. Sambas, S. Vaidyanathan, M. Mamat, M.A. Mohamed, W.S. Mada Sanjaya, A new chaotic system with a pear-shaped equilibrium and its circuit simulation. Int. J. Electr. Comput. Eng. 8, 4951–4958 (2018) 29. M.A. Savi, F.H.I. Pereira-Pinto, F.M. Viola, A.S. de Paula, D. Bernardini, G. Litak, G. Rega, Using 0-1 test to diagnose chaos on shape memory alloy dynamical systems. Chaos Solitons Fractals 103, 307–324 (2017) 30. J.P. Singh, B.K. Roy, Coexistence of asymmetric hidden chaotic attractors in a new simple 4-D chaotic system with curve of equilibria. Optik 145, 209–217 (2017) 31. R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, Austria, 2018) 32. The MathWorks, Inc., Matlab, Natick (Massachusetts, United States, 2015) 33. J. Tomˇcala, Acceleration of time series entropy algorithms. J. Supercomput. 75, 1443–1454 (2018) 34. J. Tomˇcala, TSEntropies: Time Series Entropies. R package version 0.9. (2018), https://CRAN. R-project.org/package=TSEntropies. Cited 15 Mar 2019 35. S. Vaidyanathan, A. Sambas, S. Kacar, Ü. Çavu¸so˘glu, A new three-dimensional chaotic system with a cloud-shaped curve of equilibrium points, its circuit implementation and sound encryption. Int. J. Model. Identif. Control 30, 184–196 (2018) 36. S. Vaidyanathan, A. Sambas, M. Mamat, A new chaotic system with axe-shaped equilibrium, its circuit implementation and adaptive synchronization. Arch. Control Sci. 28, 443–462 (2018)

Part II

Applications of Nonlinear Dynamics

Chapter 6

Predictability and Entropy of Supercomputer Infrastructure Consumption Jiˇrí Tomˇcala

Abstract This paper examines the relationship between time series predictability and development of its entropy. It describes and graphically illustrates the principle of both algorithms and derives their logical connection based on this. This deduced hypothesis is then tested on an artificially created time series as well as on the real-world time series. The results are then systematically and clearly graphically displayed and evaluated.

6.1 Introduction Knowing predictability is important in many areas. For example, biology studies the predictability of human behavior depending on the genetic code. In macroeconomics, predictability determines to what extent real data corresponds to the economic model. There is also so-called spring predictability barrier phenomenon in the weather forecast that indicates the unusual difficulty of summer whether prediction about the El Niño–Southern Oscillation. An interesting example is quantum mechanics, where predictability is strongly limited by the Heisenberg’s indeterminacy principle. It can be said that predictability is a degree to which a correct prediction can be made. This attribute suggests how much it makes sense to attempt any prediction. Predictability can be determined by performing a series of predictions and comparing them with reality, but this way is really time-demanding. The aim of this work is to propose the faster way, which uses entropy calculation of analyzed time series to determine its predictability. Similarity of prediction algorithm and algorithm for calculation of entropy will be shown in this work in section Algorithms comparison. Last but not least, a comparison with the development of a chaos test is added in numerical experiments.

J. Tomˇcala (B) IT4Innovations, VŠB—Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_6

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6.2 Prediction Methods First, it is appropriate to write a few words about the existing prediction methods. Nowadays, artificial intelligence is involved in solving various problems. Machine learning methods are one of its offshoots and can be used, inter alia, also to predict time series. The basic idea is to build a training set and then teach it an abstract model. That is called supervised learning. The mechanism of assembling the training set from the predicted time series is shown in the following lines. Let x denote the predicted time series and X assembled training set. Then x = (x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , . . .)

(6.1)

⇓ Observation Predictor Predictor Target x(1) = (x1 , x2 ) y (1) = x3

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(6.2)

For calculation of time series predictions can be used machine learning methods such as linear and polynomial regression, random forest, k-nearest neighbors or artificial neural networks, which all can be applied to this regression problem. Another approach is to use nonlinear prediction algorithms such as zeroth-order algorithm or radial base function. Their principles are described here [1]. The oldest approach is statistical. The most used statistical methods for predictions are simple moving average (SMA), exponential smoothing (ETS), and probably the best known auto-regressive integrated moving average (ARIMA). Detailed information about these methods can be found here [2].

6.3 Entropies and Chaos Generally, entropy is a measure of the disorder. Time series entropy is also often the degree of time series complexity. There are several algorithms for calculating time series entropy, the most widely used of which are Approximate and Sample Entropy. To quickly determine the entropy of the time series, it is appropriate to use Fast Approximate Entropy, which is an accelerated modification of Approximate Entropy. This modification was proposed in [3] and its speedup is of the order of 103 .

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As for the notion of chaos, it can be said that the time series is chaotic if there are no signs of regular behavior. Such a time series should then be completely unpredictable. For the detection of chaotic dynamics of analyzed time series is in this work used so called 0-1 test for chaos [4]. The result of this test can only be the statement that the analyzed time series is chaotic (the test value is close to number 1), or that this analyzed time series is regular (the test value is close to 0). If the value is somewhere between 0 and 1, this test is not able to distinguish between chaotic and regular patterns.

6.4 Algorithms Comparison In this pivotal section there is a comparison of k-nearest neighbors (k-nn) prediction algorithm and the algorithm for calculation of Fast Approximate Entropy (FastApEn). The purpose of this comparison is to find out how much these algorithms are similar and whether a similar development of entropy and accuracy of time series predictions can be expected at all. The similarity of the principles of the two algorithms can be seen at first glance from Figs. 6.1 and 6.2. Also, common features in (6.3) and (6.4) can be observed to calculate k-nn prediction and FastApEn value. The k-nn prediction accuracy, as well as the FastApEn value, depends on the density of the points in the analyzed neighborhood of the training set resp. sub-sequences space. It is quite likely that if close observations are found in the k-nn training set, then also the prediction will be close to the actual value. Likewise, in FastApEn, when searching for close sub-sequences, their density in the near vicinity of the analyzed neighborhood causes a decrease in the entropy value. When the meanings of the previous sentences are combined, then through transitivity it can be assumed that the k-nn prediction accuracy will be much higher with a lower FastApEn value. This important assumption is validated in the following chapter, where the results of numerical experiments are presented. The above mentioned prediction method is introduced by: 1 zi , k i=1 k

yˆ = KNNR(X, x , k) =

(6.3)

z = {y (ξ ) : x(ξ ) ∈ U (x ) ∧ |U (x )| = k ; ξ ∈ 1, N }, where X is the set of training examples, x is the time series whose development is predicted, k is the number of nearest neighbors, N is the number of training examples, KNNR() is the k-nearest neighbors regression function, and yˆ is the prediction itself. The Fast Approximate Entropy used in this paper is defined in the following way:

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Fig. 6.1 The principle of the k-nearest neighbors prediction algorithm. Each dot represents one observation of training set. The algorithm looks for k nearest observations from the training set to the analyzed time series. The forecast of this analyzed time series is then calculated as the arithmetic mean of the targets of these k closest observations. The neighbourhoods are defined most often by the Euclidean norm and are thus represented in this figure as blue circles

FastApEn(x, m, r ) = N  Nm+1     m  1  |ci,m | |ci,m+1 | 1 = log log − , Nm i=1 Nm Nm+1 i=1 Nm+1

(6.4)

/ c j,m , j < i}, yi = [xi , xi+1 , . . . , xi+m−1 ], ci,m = {ξ | yi − yξ  ≤ r ∧ ξ ∈ where ci,m are value classes of subsequences of length m and Nm is the number of these value classes.

6.5 Numerical Experiments To verify the above assumption, an artificially created time series testTS: testTS(t) = sin(t) +

rnormTS(t) − sin(t) , 1 − e−t

(6.5)

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Fig. 6.2 The principle of the Fast Approximate Entropy algorithm. Each dot represents one subsequence of length m = 2. The blue dots are subsequences, in the neighborhood of which further subsequences are searched for. The blue squares then illustrates these neighborhoods. These are areas where the maximum norm of distance from blue dots is less than r . The black dots outside of squares are subsequences not classified yet. Inside of squares are already classified. Each blue square represents one value class

where rnormTS is the random time series with a normal probability distribution, was used, as was the real-world time series—the normalized measured input power from IT4Innovations supercomputer infrastructure. All k-nn predictions were calculated using R [5] package caret [6] and FastApEn values were calculated using R package TSEntropies [7]. Due to an interesting comparison, the maximal Lyapunov exponent development was calculated using R package nonlinearTseries [8] and also development of 0-1 test for chaos using R package Chaos01 [9]. Also, for the sake of clarity in the development of the prediction error, the RMSE (Root Mean Squared Error) value is added. The development of all investigated values is clearly shown in Figs. 6.3 and 6.4.

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Fig. 6.3 Comparison of developments of test time series, k-nn method prediction error, RMSE, Fast Approximate Entropy, Maximal Lyapunov exponent and the 0-1 test for chaos

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Fig. 6.4 Comparison of normalized measured input power from IT4Innovations supercomputer infrastructure, k-nn method prediction error, RMSE, Fast Approximate Entropy, Maximal Lyapunov exponent and the 0-1 test for chaos. The measured time period is from 1 A.M., November 2 until 9 P.M., November 5, 2017

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6.6 Conclusion From the graphical comparison of the course of individual values it can be said that the assumed relationship between entropy and prediction error was confirmed. Thus, the entropy value can be demonstrably taken as a plausible guide in estimating the predictability of time series. An interesting fact is that this relationship was in real-world time series case traced despite the result of 0-1 test for chaos. That is because it moves very close to the value 1 and thus, according this test, this time series is chaotic all the time. It can be seen from the development of the maximal Lyapunov exponent that its value could also be used as a guideline for estimating the predictability of the time series, but the development of the FastApEn value appears to be much more explicit. Acknowledgements This work was supported by The Ministry of Education, Youth and Sports of Czech Republic from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center—LM2015070”.

References 1. H. Kantz, T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, New York, 2003) 2. A. Bovas, J. Ledolter, Statistical Methods for Forecasting (Wiley-Interscience, 2013) 3. J. Tomˇcala, Acceleration of time series entropy algorithms. J. Supercomput. 75, 1443–1454 (2018) 4. G.A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 460(2042), 603–611 (2004) 5. R Core Team, R: A Language and Environment for Statistical Computing (R Foundation for Statistical Computing, Vienna, Austria, 2018). Accessed 7 Jun 2019. https://R-project.org 6. M. Kuhn, caret, R package, CRAN repository. Accessed 7 Jun 2019. https://CRAN.R-project. org/package=caret 7. J. Tomˇcala, TSEntropies, R Package, CRAN repository. Accessed 7 Jun 2019. https://CRAN.Rproject.org/package=TSEntropies 8. C.A. Garcia, G. Sawitzki, NonlinearTseries, R Package, CRAN repository. Accessed 7 Jun 2019. https://CRAN.R-project.org/package=nonlinearTseries 9. T. Martinoviˇc, chaos01, R Package, CRAN repository. Accessed 7 Jun 2019. https://CRAN.Rproject.org/package=Chaos01

Chapter 7

Chaotic Approach Based Feature Extraction to Implement in Gait Analysis ˙ S. Ikizo˘ glu and B. Atasoy

Abstract Chaotic systems, which are fundamentally described as the systems having sensibility to the initial conditions, are extensively used in a broad area ranging from economics to engineering sciences, from biomedical to space researches. There exist methods to analyze chaos and gives an estimation, which are very helpful for dynamic analyses of systems being nonlinear or cannot be modeled mathematically based on differential equations. This study focuses on extracting features based on nonlinear time series to describe problems leading to balance disorder. For this purpose, required data is collected from pressure sensors placed at certain locations under the sole. A two-step classification process is projected, where the first step will determine the subject to belong either to ‘healthy’ group or to ‘ill’. In case the latter is assigned, a second step will follow where we search for the ability of chaotic approach to provide features to define the specific illness. During the study, firstly the delay coordinates have been constructed form the obtained time series, hence the nonlinear time series were hardly difficult to construct a suitable mathematical modelling [1]. Thereafter, optimal time delay and embedding dimensions [2–4] have been chosen, and the delay coordinates’ analysis for the shape of attractors [3], Poincare Mapping [3], and Fractal Analyses [3, 5] have been examined in order to detect a feature to distinguish the type of illness.

7.1 Introduction Chaos is fundamentally described as the irregularity in the regular systems, which has a broad range of area of application such from economics to engineering, or medical to aerospace sciences.

S. ˙Ikizo˘glu Department of Control & Automation Engineering, Istanbul Technical University, Istanbul, Turkey B. Atasoy (B) Department of Mechanical Engineering, Pîrî Reis Maritime University, Istanbul, Turkey e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_7

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Chaos theory in dynamical systems generally deals with the systems having partial of full nonlinearity. As mentioned before, these irregularities mostly collapses with the “Superposition Principle”, which also corresponds the system to have a nonlinear characteristics. Additionally, sometimes a mathematical model is hardly able to be constructed for some chaotic systems, so a coordinate system for mth order metric space is employed to analyse. In this study, a chaos analysis for gait analyses has been performed. There exists a theoretical approach to implement whether one may distinguish the healthy people from patients, and then any feature has been examined to extract the kind of balance disorders. So, in the first part, a theoretical background and some studies on literature has been explained. In the second part, experimental mechanism and data analyses processes has been performed. In the third part, the chaotic analyses methods have been explained and the data related with healthy people and patients have compared, and as well as the results have been discussed in the fourth part.

7.1.1 Literature Survey Chaos theory is widely used to simulate the biomedical studies, such include the Electroencephalography (EEG), Heart Rate Variability (HRV) monitoring, and as well as gait analysis. Marmelat, V. and Meidinger, R. L. (2019) have studied the fractal analyses methods for Parkinson diseases, by using alpha-Detrended Fluctuation Analyses (α-DFA). They firstly performed a 3-min-analyses as 5 times, and then a long time analysis of 15 min was done. During the experiment, they have noticed that short analyses do not enable to estimate the characteristics for the long measurement of Parkinson disease. Additionally they have also concluded that 15 min analysis is too long for a patient to distinguish the characteristics [6]. Xia, Y. (2015) has examined to find another way to distinguish the type of Neurodegenerative Diseases (NDD). They performed some statistical and entropy analyses methods based on neuro-fuzzy models. Such analyses tools are skewness, LempelZiv Complexity, curtosis and Teager-Kaiser energy methods. During their study, satisfactory results for ALS and Parkinson’s diseases have been obtained [7]. Sarbaz, Y., Towkidhah, F., Jafari, A. and Gharibzadeh, S. (2012) have studied to analyse the Parkinson’s disease using chaos theory. They firstly have obtained the time series from patients of Parkinson’s disease. In order to perform this, firstly the metric space has been reconstructed, and then the three maximal Lyapunov exponents have been analyzed. During the study, they have noticed that the chaotic dynamics of the Parkinson’s disease have altered and there exists a stochastic characteristic for the disease rather than deterministic chaos [8].

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7.1.2 Theoretical Background As mentioned in the previous part, mostly Parkinson’s disease were under examination as a balance disorder in the literature. In this study, we proposed a new method to distinguish another type of balance disorders by using chaotic analyses method. Theoretically, a step can be modeled as a function of time, so a time series analysis is enabled. In order to model the step, force analysis is a very useful method. So in the study, Piezo Resistive Force (PRF) sensors have been employed. After the data acquisition process has been completed, the first part of the analysis has been proved. According to this approach, although the frequencies and the amplitudes of the time series are different, the shape should be relevant to each other for the healthy people. So that, the shape of attractors in 3D phase space should be similar. However, in case for the patients, the time series data must have anomalies, such as some unique frequency and amplitude values. So, by using the Delay Coordinates Method, shape of the attractors should be as different as healthy people, which can be a simple and effective tool to distinguish the healthy and patient people.

7.2 Experimental Mechanism In the study, PRF sensors have been employed. As an arrangement, 8 sensors have been placed under the sole with an optimal placement as given below [9] (Fig. 7.1). During the data acquisition, the sensor gives analog voltage values as input, so a data conversion process is required. The characteristic result has been obtained by using Curve Fitting Method, and the sensor output characteristic equation is given as:

Fig. 7.1 Arrangement of the PRF sensors (Left) and voltage-to-weight calibration curve (Right) [9]

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y = exp((x + 0.2245)/0.9265)

(7.1)

Then the sensors have been connected to Arduino Mega 2560 via a Bluetooth module, and the time series have been generated by acquiring the 8-channel PRF sensor data independently. After obtaining the 8-channel time series, a MATLAB code that runs TISEAN software codes. The MATLAB code have separated and examined each channel data independently, and then given the required plots.

7.2.1 Delay Coordinates Method for Healthy and Patients As a first part of the analysis, 4 healthy people’s step data were analysed to prove the first part of the analyses. To perform the delay coordinates, two important parameters of Optimal Delay Time has been determined via Average Mutual Information Function [3], and the embedding dimension values have been obtained via False Nearest Neighborhood methods [4]. The results for the healthy people is given as below [9] (Fig. 7.2). And the shape of the attractors for the healthy and patient people of the same illness group is plotted below [9] (Fig. 7.3).

Fig. 7.2 Shape of attractors constructed by the healthy people’s time series data obtained by A5 channel [9]

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Fig. 7.3 Shape of attractors’ comparison for a healthy person and three patients of Meniere syndrome for A6 channel [9]

7.3 Results and Discussion In this study, gait analyses of step data has been examined via chaotic analysis methods. In order to perform that, 8 channel PRF sensors have been placed to the underneath of the feet, and the related force data have been obtained as time series. After the data acquisition, the nonlinear time series data have been analyzed under MATLAB and TISEAN environments together. In the first part, healthy people’s attractor shapes, Poincare Maps and Information Dimensions have been examined, and thereafter these procedures have been repeated for the patient groups to extract a feature that distinguishes the type of illnesses. During the analyses part, the shape of attractors have been found as similar for the healthy people, whereas in the case for patient groups, the shape of attractors have been examined as quietly different form healthy people. This gives that, it is possible to distinguish the healthy and patient people by using their attractor shapes. However, during the analyses steps, Poincare Maps and Information Dimensions have given no relevant results, so different analyses methods may be employed to extract the kind of illnesses.

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References 1. J. Gao, Y. Cao, W.W. Tung, J. Hu, Chaotic time series analysis, in Multiscale Analysis of Complex Time Series (Wiley, New Jersey, 2007), p. 243 2. J. Gao, Y. Cao, W.W. Tung, J. Hu, Chaotic time series analysis, in Multiscale Analysis of Complex Time Series (Wiley, New Jersey, 2007), p. 235 3. D. Yılmaz, F. Güler, Kaotik Zaman Serisi Üzerine Bir Ara¸stırma. J. Fac. Eng. Arch. Gazi Univ. 21(4), 775–779 (2006) 4. M. Kennel, R. Brown, H.D.I. Abarbanel, Determining embedding dimension for phase-space reconstruction using a geometrical construction. Phys. Rev. A 45(6), 3403–3411 (1992) 5. J. Gao, Y. Cao, W.W. Tung, J. Hu, Chaotic time series analysis, in Multiscale Analysis of Complex Time Series (Wiley, New Jersey, 2007), p. 251 6. V. Marmelat, R.L. Meidinger, Fractal analysis of gait in people with Parkinson’s disease: three minutes is not enough. Gait Posture (2019) 7. Y. Xia, Q. Gao, Classification of gait rhythm signals between patients with neuro-degenerative diseases and normal subjects: experiments with statistical features and different classification models. Biomed. Signal Process. Control (2015) 8. Y. Sarbaz, F. Towhidkhah, Do the chaotic features of gait change in Parkinson’s disease? J. Theor. Biol. (2012) 9. B. Atasoy, Chaotic time series based feature extraction to use for determination of balance disorders published master’s thesis (Istanbul Technical University, Istanbul, Turkey, 2019)

Chapter 8

Characterization of Cardiac Cell Electrophysiology Model Using Recurrence Plots Radek Halfar

Abstract The main aim of this paper is to analyse the evolution of cardiac cell transmembrane potential forced by periodic pacing. For this purpose, the BeelerReuter model of ventricular cardiac cell is used. The Beeler-Reuter model is a well stated mathematical model of a cardiac ventricular cell. Many papers dealing with heart electrophysiology using this model for its great properties. In this paper, the model is forced by pacing stimulus with the shape of the half-sine period followed by zero function. The computed model motions are investigated using the recurrence plots, recurrence quantification analysis, and approximate entropy with respect to the pacing period.

8.1 Introduction Since heart diseases are the most common cause of death in the world [5, 19] it is very important to understand the proper heart work. The mechanical heart work is governed by electrical impulses generated by the heart itself. These impulses propagate through the heart and make the heart pump blood into the body. Therefore on the proper propagation of these impulses depends on each individual’s life. In this paper, the recurrence plots and recurrence quantification analysis (RQA) are used for investigation of dynamic of the Beeler-Reuter cardiac cell model of transmembrane potential and the influence of the pacing period to this dynamic. Since its introduction in 1987 by Eckmann et al. [7] recurrence plot (RP) became an important tool for the investigation of the dynamical system. Since then, RPs were successfully used for authentication in the Internet of Things [2], automatic Parkinson’s disease identification [1], investigation of heart rate variability [16] and many others.

R. Halfar (B) IT4Innovations, VŠB—Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_8

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8.2 Beeler-Reuter Model The Beeler-Reuter model of the cardiac cell proposed by Beeler and Reuter in 1977 [3]. Model is established by eight Equations (8.1) defining the time derivatives of transmembrane potential Vm in mV, intracellular Ca 2+ concentration [Ca]i in mole/l, and six dimensionless gating variables x1 , m, h, j, d, and f . Gating variables are in (8.1) modelled as variable y (difference in equations for the particular gating variables is given by constants). i ext − i k1 − i x1 − i N a − i Ca d Vm = , dt Cm d[Ca]i = −10−7 i s + 0.07(10−7 − [Ca]i ), dt dy y∞ − y . = dt τy

(8.1)

Times t and τ are in ms. Parameter Cm defines membrane capacitance in µF/cm2 (in this study Cm = 1). Detailed information about the Beeler-Reuter model can be seen in [3]. Typical ventricular transmembrane potential computed using Beeler-Reuter model can be seen in Fig. 9.1. In this paper, the externally applied current i ext are impulses with a duration of 1 ms and amplitude of 80 µA/cm2 created by the first half period of the sine function followed by the zero function. Variable i ext is therefore defined by the following equation:  i ext =

80 sin(π(t − n(c + 1))) 0

t ∈ [n(c + 1), n(c + 1) + 1], t∈ / [n(c + 1), n(c + 1) + 1].

The graphical representation of variable i ext can be seen in Fig. 8.2. Fig. 8.1 Transmembrane potential (variable Vm ) of stimulated ventricular cardiac cell computed using Beeler-Reuter model

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The model equations were solved numerically using the variable order solver based on the numerical differentiation formulas implemented as an ode15s solver in MATLAB [17]. The computations were performed for the stimulation delays c from 10 to 365 ms with a step of 5 ms. Each simulation was done for the time from 0 to 105 ms. Subsequent analysis of resulting transmembrane potential using RP was performed in R [18] designed by R Core Team using packages nonlinearTseries [8] and fractal package [6].

8.3 Main Results In order to exclude transient phenomena, the only transmembrane potential (variable Vm ) from 90 × 103 to 105 ms were analyzed. From this time period, the values for subsequent analysis were selected with 1 ms time step. The investigated time series is thus a length of 10001 data points.

8.3.1 Recurrence Plots From this time series, the time delay was firstly estimated. This estimation was computed using fractal package [6] by the first zero crossing of the autocorrelation function. Next, the number of embedding dimension was estimated by the algorithm suggested by Cao [4] and calculated using nonlinearTseries package [8]. The threshold for defining to states as a recurrence was selected as a 3% of phase space diameter. Next, the recurrence plots were computed using nonlinearTseries package. Resulting RPs can be divided into two groups. One group consists of RP made by long

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In the second case, the diagonals are interrupted (c ∈ {30} ∪ [45, 50] ∪ [65, 70] ∪ [130, 140]). Examples of these RPs can be seen in Figs. 8.6 and 8.7. These lines represent time intervals, where the trajectory in the phase space runs parallel to another sequence of this trajectory (dynamics is similar). In RPs can also be seen certain vertical distances, but these distances are not as regular as in the previous case. In several cases in this group can be seen a small rectangular patch which

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Fig. 8.7 Resulting reccurence plot (Left) and transmembrane potential (right) for c = 100 ms

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rather looks like the RP of the periodic motion (see Fig. 8.7). This structure reveals an unstable periodic orbit.

8.3.2 Recurrence Quantification Analysis Next, the recurrence quantification analysis (RQA) was computed. RQA is a method that belongs to nonlinear data analysis. This technique quantifies the number and duration of recurrences of a dynamical system in state space. Details about RQA can be founded in [15]. These calculation were performed in R using nonlinearTseries package [8]. The resulting RPs were analyzed using several measures. The best results were achieved by calculating the length of the longest diagonal line (L max ) and the ratio between the percentage of diagonal lines in the RP DET, and density of recurrence points in a recurrence plot RR. Results of RQA can be seen in Fig. 8.8. In this figure can be seen that the length of the longest diagonal line can be divided into two groups. One group consist of RPs with L max ≤ 8000. This RPs can be seen for c ∈ {25, 30, 45, 50, 65, 70} ∪ [90, 100] ∪ [130, 140], and the RP with L max > 8000 which was observed elsewhere. The measure ratio divides the RP into two groups as well. For the RP with ratio ≥ 500 for the stimulation delays c ∈ {25, 45, 50, 65, 70} ∪ [90, 100] ∪ {130, 135} and for ratio < 500 observed elsewhere. Notice, that results of RQA correspond to the observations made using RPs.

8.3.3 Approximate Entropy

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Next, the approximate entropy (ApEn) was computed. This technique allows comparing the complexity of the system with different parameters setting. Another advantage of this method is that it can be calculated in a short time series. For more details about approximate entropy see [13, 14]. The calculations of this technique were performed in R using the TSEntropies package. Neighborhood threshold r was defined as a 10% of phase space diameter.

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The ApEn of Beeler-Reuter model can be seen in Fig. 8.9. In this figure can be seen an increase of complexity around stimulation delay 30, 45, 70, 90, and 135 ms. Local extrema on similar values of c can be seen also in Fig. 8.8. In this figure can be seen that results of RQA (Fig. 8.8) and ApEn (Fig. 8.9) coincide for most values of parameter c.

8.4 Conclusions

0.20 0.10 0.00

ApEn

0.30

In this paper, the transmembrane potential calculated using the Beeler-Reuter model was analyzed with respect to the stimulation period. The model (8.1) was paced by the stimulus with the shape of the half-sine period. For the solving model equations, variable order solver based on the numerical differentiation formulas implemented as an ode15s solver in MATLAB was used. It was observed, that with periodical forcing the model shows periodic as well as non-periodic motions and the complexity of data vary with stimulation period. For the evaluation of motion, the recurrence plots, recurrence quantification analysis (see Fig. 8.8), and approximate entropy (see Fig. 8.9) were used. The achieved results can be compared to the investigation of the Beeler-Reuter model using the 0-1 test for chaos published in [11]. This test was designed to distinguish regular and chaotic dynamics. The resulting value of this test close to 0 shows regular behaviour, and result close to 1 shows chaotic motion. For more detailed description of this test see [9, 10]. Comparing Figs. 8.8 and 8.10 can be seen, that the result of RQA and the 0-1 test for chaos coincide in most values. The difference (the 0-1 test for chaos indicates regular motion and the ratio ≥ 500 or L max ≤ 8000 and vise versa) in these figures can be found for c ∈ {25, 30, 50, 100} for RQA measure L max , and for c ∈ {25, 50, 100, 140} for RQA measure ratio. The achieved results can be also compared with paper [12], where dynamical properties of modified Fenton-Karma model of the heart cell was investigated.

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Acknowledgements This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science – LQ1602”; by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center – LM2015070”; by SGC grant No. SP2019/125 “Qualification and quantification tools application to dynamical systems”, VŠB - Technical University of Ostrava, Czech Republic.

References 1. L.C. Afonso, G.H. Rosa, C.R. Pereira, S.A. Weber, C. Hook, V.H.C. Albuquerque, J.P. Papa, A recurrence plot-based approach for parkinson’s disease identification. Futur. Gener. Comput. Syst. 94, 282–292 (2019). https://doi.org/10.1016/j.future.2018.11.054, http://www. sciencedirect.com/science/article/pii/S0167739X18322507 2. G. Baldini, R. Giuliani, F. Dimc, Physical layer authentication of internet of things wireless devices using convolutional neural networks and recurrence plots. Internet Technol. Lett. (2018). https://doi.org/10.1002/itl2.81 3. G.W. Beeler, H. Reuter, Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. 268(1), 177–210 (1977). https://doi.org/10.1113/jphysiol.1977.sp011853 4. L. Cao, Practical method for determining the minimum embedding dimension of a scalar time series. Phys. D Nonlinear Phenom. 110, 43–50 (1997). https://doi.org/10.1016/S01672789(97)00118-8 5. Centers for Disease Control and Prevention, Leading causes of death (2017), https://www.cdc. gov/nchs/fastats/leading-causes-of-death.htm. Accessed 29 May 2019 6. W. Constantine, D. Percival, Fractal: a fractal time series modeling and analysis package (2017), https://CRAN.R-project.org/package=fractal. R package version 2.0-4 7. J.P. Eckmann, S.O. Kamphorst, D. Ruelle, Recurrence plots of dynamical systems. Eur. Lett. (EPL) 4(9), 973–977 (1987). https://doi.org/10.1209/0295-5075/4/9/004 8. C.A. Garcia, NonlinearTseries: nonlinear time series analysis (2018), https://CRAN.R-project. org/package=nonlinearTseries. R package version 0.2.5 9. A. Gottwald, I. Melbourne, A new test for chaos in deterministic systems. Proc. R. Soc. Lond. A 460, 603–611 (2004) 10. A. Gottwald, I. Melbourne, On the implementation of the 0–1 test for chaos. SIAM J. Appl. Dyn. 8, 129–145 (2009)

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11. R. Halfar, Dynamical properties of Beeler-Reuter cardiac cell model with respect to stimulation parameters. Int. J. Comput. Math. (to appear) 12. R. Halfar, M. Lampart, Dynamical properties of the improved FK3V heart cell model. Math. Methods Appl. Sci. 41, 7472–7480 (2018). https://doi.org/10.1002/mma.5060 13. K.K.L. Ho, G.B. Moody, C.K. Peng, J.E. Mietus, M.G. Larson, D. Levy, A. Goldberger, Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics. Circulation 96, 842–848 (1997). https://doi.org/10.1161/01.CIR.96.3.842 14. M. Lampart, T. Martinoviˇc, Chaotic behavior of the CML model with respect to the state and coupling parameters. J. Math. Chem. (2019). https://doi.org/10.1007/s10910-019-01023-2 15. N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Recurrence plots for the analysis of complex systems. Phys. Rep. 438(5), 237–329 (2007). https://doi.org/10.1016/j.physrep.2006.11.001, http://www.sciencedirect.com/science/article/pii/S0370157306004066 16. N. Marwan, Wessel, N., Meyerfeldt, U., Schirdewan, A., Kurths, J.: Recurrence-plot-based measures of complexity and their application to heart-rate-variability data. Phys. Rev. E 66, 026,702 (2002). https://doi.org/10.1103/PhysRevE.66.026702 17. Matlab (The MathWorks Inc, Natick, Massachusetts, United States, 2016) 18. R.C. Team, R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2018). https://www.R-project.org/ 19. World Health Organization, The top 10 causes of death (2017). http://www.who.int/ mediacentre/factsheets/fs310/en/. Accessed 10 May 2018

Chapter 9

Effects of Age and Illness to the Complexity of Human Stabilogram Radek Halfar

Abstract The main aim of this paper is to analyse the effects of health and age to human stability. For this purpose, the dataset with a recording of force platforms from subjects undergoing stabilography tests is utilized. These tests are consist of steady standing for 60 s. These tests were done upon several conditions and each test was repeated several times. From these recording, the approximate entropy was computed and the results were analysed using tools of mathematical statistics.

9.1 Introduction The human balance is one of the basic functions, that the human brain needs to perform. For this purpose, a large number of human organs (such as eyes, muscles, brain, etc.) need to work precisely and synchronously. When the system providing this function begins to degrade (or is damaged), stability is lost, which can lead to falls or other dangerous situations. It is therefore important to pay attention to these changes, to learn to recognize and quantify them. One of the main tools of how to examine human stability is an assessment of balance by investigating data measured by the force plate. As an example can be used paper [10] where the centre of pressure fluctuation from sitting and standing position are investigated. Another example is paper [2]. In this review paper, the strengths and limitations of measuring the centre of pressure to the multiscale entropy are calculated. In paper [3] are the centre of pressure data used to quantify balance using multivariate multiscale entropy. The effect of entropy measure parameters to the human balance evaluation is presented in [8]. In this paper, the data measured during static posturography are investigated in the means of signal complexity in order to determinate the effects of age and illness to the result of approximate entropy (ApEn) computation. For this purpose, the R. Halfar (B) IT4Innovations, VŠB—Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava, Czech Republic e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_9

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calculation of the ApEn in R using the TSEntropies package [12] is performed. Statistical computation is calculated in R [9] as well by packages ggpubr [5], moments [6], dplyr [14], tidyr [15], and ggplot2 [13].

9.2 Materials and Methods 9.2.1 Dataset In order to analyze the above-mentioned effects to the balance complexity, the Human Balance Evaluation Database (HBEDB) is used. This dataset is provided by Santos and Duarte [11] and is publicly available from PhysioNet website [1]. The HBEDB contains measurements of subjects undergoing posturography test using a force platform. One posturography test consists of one minute long steady standing on the force plate. This test has been done under four different conditions. • • • •

Open eyes on firm surface. Open eyes on foam surface. Closed eyes on firm surface. Closed eyes on foam surface.

The overall number of tested subjects is 163 and every condition is tested 3 times (total number of measurements in the dataset is 1930). The sampling frequency is 100 Hz and for filtering, the low pass filter is used (cut off frequency of the filter is 10 Hz). Each recording is made of 8 different time series. • Force (N) measured in x, y, and z. • Moment of forces (Nm) measured in x, y, and z. • Center of pressure (cm) calculated in x, and y. The dataset was divided according to two different rules. One division of the dataset is according to the age of subject to young (age < 60) and old (age ≥ 60). The second rule is whether the subject has any illness (hypothyroidism, asthma, dermatitis, diabetes mellitus etc.), as declared by themselves (yes or no). For more information about HBEDB dataset reader can see [11].

9.2.2 Approximate Entropy The approximate entropy (ApEn) is a technique for quantifying the complexity of the system. The advantages of ApEn are lower computational demand, possibility to use this method for a short time series and less effect from noise. For more detailed information about ApEn see [4, 7].

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9.2.3 Approximate Entropy Calculation As a first step of analyzing data from dataset, a new time series was created from the measurements. This new time series (force F, moment of force MoF and centre of pressure CoP) was calculated as a module from the axes of the investigated system (see 9.1).  F = FX2 + FY2 + FZ2  MoF = MoFX2 + MoFY2 + MoFZ2  CoP = CoPX2 + CoPY2 + CoPZ2

(9.1)

Using these new variables, the ApEn was calculated in R using the TSEntropies package. Neighbourhood threshold r was defined as a 0.2σ of investigated time series and dimension of a given time series was defined as 2. Next step was to calculate from all tests performed by one subject the mean value of ApEn for the analyzing system (F, MoF, CoP) of each tested subject. Therefore one tested subject is represented by three values ( ApEn F , ApEn MoF , and ApEn CoP ). In Figs. 9.1 and 9.2 can be seen boxplots of resulting values of ApEn. In these figures can be seen that values of ApEn computed from variables MoF and CoP

Fig. 9.1 Boxplots of ApEn depicted according to the illness of subject

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has similar variance and mean value. ApEn computed from variables MoF and CoP reaches lower values with a group of the young and healthy subject over the old and ill people. On the other hand, calculated values of ApEn from variable F reaches greater values and young and healthy people revealing the greater value of ApEn over the old and ill subjects. For subsequent analysis, outliers are removed from the investigated dataset.

9.3 Statistical Analysis Next step was to test whether the data follows the normal distribution. For this purpose, the Shapiro–Wilk test was applied. In this test, the null hypothesis is that the data are normally distributed. Thus, if the p-value is less than the chosen α level (in this paper α = 0.05), the null hypothesis is rejected (and tested data are not normally distributed). The result of this tests can be seen in Tables 9.1 and 9.2. In these tables can be seen, that almost all variables did not come from the normal distribution. Due to the rejection of the null hypothesis of the Shapiro–Wilk test, the MannWhitney U test was used to test the effects of age and illness on the complexity of the measured data. It is a non-parametric test that is used to test whether two medians from independence samples are equal or not. The null hypothesis of this test is that the medians from two independent samples are equal. The results of these tests can be seen in Table 9.3. In this table can be seen, that the null hypothesis is rejected in all tested cases. This fact implies that there is a statistically significant difference between the medians of ApEn between healthy and ill people and between young and old subjects.

Table 9.1 Results of Shapiro-Wilk tests ( p-values) computed from ApEN according to illness Type Yes No MoF F CoP

0.167 < 0.001 0.516

0.016 0.014 0.008

Table 9.2 Results of Shapiro-Wilk tests ( p-values) computed from ApEN according to age Type Young Old MoF F CoP

0.019 0.019 0.005

0.211 0.014 0.456

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Table 9.3 Results of Mann-Whitney U tests ( p-values) computed from ApEN according to illness and age Type Illness Age MoF F CoP

0.027 0.002 0.014

< 0.001 < 0.001 < 0.001

9.4 Conclusion In this paper, the effects of illness and age to the complexity of data acquired during posturography tests are investigated. For this purpose, the approximate entropy from the acquired force, the moment of forces, and the centre of pressure are computed. The effect of age (young and old) and illness (hypothyroidism, asthma, dermatitis, diabetes mellitus etc.) was subsequently investigated using statistical analysis. Using boxplots has been shown that approximate entropy is greater for the moment of forces and centre of pressure measured with ill subjects and measured force with healthy subjects. The greater complexity was also measured in force of young people and MoF and CoP of old subjects. The statistically significant effects are also confirmed by hypothesis testing. These results should be kept in mind by all researcher dealing with posturography data because the results of studies dealing with the complexity of stability can be affected by many factors. When preparing human stability experiments, detailed information on tested subjects, such as age and health, should be recorded. Acknowledgements This work was supported by The Ministry of Education, Youth and Sports from the National Programme of Sustainability (NPU II) project “IT4Innovations excellence in science – LQ1602”; by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental Development and Innovations project “IT4Innovations National Supercomputing Center – LM2015070”; by SGC grant No. SP2019/125 “Qualification and quantification tools application to dynamical systems”, VŠB - Technical University of Ostrava, Czech Republic.

References 1. A.L. Goldberger, L.A.N. Amaral, L. Glass, J.M. Hausdorff, P.C. Ivanov, R.G. Mark, J.E. Mietus, G.B. Moody, C.K. Peng, H.E. Stanley, PhysioBank, PhysioToolkit, and PhysioNet: components of a new research resource for complex physiologic signals. Circulation 101(23), e215–e220 (13 June 2000). Circulation Electronic Pages: http://circ.ahajournals.org/content/101/23/e215. fullPMID:1085218; https://doi.org/10.1161/01.CIR.101.23.e215 2. B. Gow, C.K. Peng, P.M. Wayne, A. Ahn, Multiscale entropy analysis of center-of-pressure dynamics in human postural control: methodological considerations. Entropy 17, 7926–7947 (2015). https://doi.org/10.3390/e17127849 3. C.W. Huang, P.D. Sue, M. Abbod, B. Jiang, J.S. Shieh, Measuring center of pressure signals to quantify human balance using multivariate multiscale entropy by designing a force platform.

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Sensors (Basel, Switzerland) 13, 10151–10166 (2013). https://doi.org/10.3390/s130810151 4. K.K.L. Ho, G.B. Moody, C.K. Peng, J.E. Mietus, M.G. Larson, D. Levy, A. Goldberger, Predicting survival in heart failure case and control subjects by use of fully automated methods for deriving nonlinear and conventional indices of heart rate dynamics. Circulation 96, 842–848 (1997). https://doi.org/10.1161/01.CIR.96.3.842 5. A. Kassambara, ggpubr: ‘ggplot2’ Based Publication Ready Plots (2018). https://CRAN.Rproject.org/package=ggpubr. R package version 0.2 6. L. Komsta, F. Novomestky, Moments: moments, cumulants, skewness, kurtosis and related tests (2015). https://CRAN.R-project.org/package=moments. R package version 0.14 7. M. Lampart, T. Martinoviˇc, Chaotic behavior of the CML model with respect to the state and coupling parameters. J. Math. Chem. (2019). https://doi.org/10.1007/s10910-019-01023-2 8. L. Montesinos, R. Castaldo, L. Pecchia, Selection of entropy-measure parameters for force plate-based human balance evaluation, in World Congress on Medical Physics and Biomedical Engineering 2018, ed. by L. Lhotska, L. Sukupova, I. Lackovi´c, G.S. Ibbott (Springer Singapore, Singapore, 2019), pp. 315–319 9. R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2018). https://www.R-project.org/ 10. M. Roerdink, P. Hlavackova, N. Vuillerme, Center-of-pressure regularity as a marker for attentional investment in postural control: a comparison between sitting and standing postures. Hum. Mov. Sci. 30(2), 203 – 212 (2011). https://doi.org/10.1016/j.humov.2010.04.005, http:// www.sciencedirect.com/science/article/pii/S0167945710000588. Basic Mechanisms underlying Balance Control under Static and Dynamic Conditions 11. D.A. Santos, M. Duarte, A public data set of human balance evaluations. PeerJ 4, e2648 (2016). https://doi.org/10.7717/peerj.2648 12. J. Tomcala, TSEntropies: Time Series Entropies (2018). https://CRAN.R-project.org/ package=TSEntropies. R package version 0.9 13. H. Wickham, ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York (2016). http://ggplot2.org 14. H. Wickham, R. François, L. Henry, K. Müller, dplyr: A Grammar of Data Manipulation (2018). https://CRAN.R-project.org/package=dplyr. R package version 0.7.7 15. H. Wickham, L. Henry, tidyr: Easily Tidy Data with ‘spread()’ and ‘gather()’ Functions (2018). https://CRAN.R-project.org/package=tidyr. R package version 0.8.2

Part III

Nonlinear Circuits

Chapter 10

Hybrid Memristor-CMOS Based FIR Filter Design K. Alammari, A. Sasi, M. Ahmadi, A. Ahmadi and M. Saif

Abstract CMOS technology appears to reach its limit in terms of scaling. Thus, Memristor devices are emerged, as a solution to overcome this problem, as data can be stored with high precision in memristor devices. This paper, presents a compact, low power design of hybrid Memristor-CMOS based Finite Impulse Response (FIR) filter, with reasonable performance compared to the CMOS based design. The proposed design has been described using Verilog High Description Language (HDL) and tested with Cadence design systems, NC-Verilog and Matlab. The simulation results have shown that the behavioral model of the design can distinguish between all input signals and passes only signals with the desired frequency. The proposed hybrid memristor-CMOS based FIR is shown to be more efficient in terms of area, consumed power and delay.

10.1 Introduction The development of CMOS transistors shows major concerns, such as, increased leakage power, reduced reliability, and high fabrication cost [1]. These factors have affected chip manufacturing procedure and functionality severely. Therefore, the demand for new devices is increasing. Memristor, is considered as one of the key K. Alammari · A. Sasi · M. Ahmadi (B) · A. Ahmadi · M. Saif (B) Department of Electrical & Computer Engineering, University of Windsor, Windsor, Canada e-mail: [email protected] M. Saif e-mail: [email protected] K. Alammari e-mail: [email protected] A. Sasi e-mail: [email protected] A. Ahmadi e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_10

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element in memory and information processing design [2, 3] due to its small size, long-term data storage, low power, and CMOS compatibility. Recently, several methods in a logic design employing Memristors have been reported [4–7]. Material Implication Logic (IMPLY) [4, 5] is aimed to design logic gates with Memristors only. However, the involvement of a high number of computational steps and the need for READ/WRITE circuit are obstacles face this logic. Memristor Aided Logic (MAGIC) [6] is another pure Memristive logic method similar to IMPL logic. This logic requires two steps “initialization procedure prior to a computational step”. However, cascading two or more logic circuits with MAGIC is difficult to implement. Memristor Ratioed Logic (MRL) [7] method is hybrid CMOS-Memristor circuit that has logical states (‘0’ and ‘1’) represented based on the level of the output voltage. This method is suitable for most logic designs because of CMOS compatibility, which offers less area and power consumption compared to conventional CMOS. There are several Memristor based arithmetic and computational circuits described [8, 9] using MRL however, the area of filter design using Memristors has received less attention. In [8] Memristors have been used in the LC filter as a damping component. In [10] Memristor devices were used to provide weights for 6-tap FIR filter. In our paper, Memristor based RRAM device has been employed to implement hybrid MemristorCMOS based FIR filter. Unlike earlier memristor design [10], the proposed approach utilizes the designed cell library. The cell library proposed characterizes basic Memristive logic gates based on MRL method. The rest of this paper is organized as follows. An introduction to the Memristor model is described in Sect. 10.1. In Sect. 10.2, the design approach and device modeling are explained. In Sect. 10.3, FIR filter design is presented, Sect. 10.4, a case study of the DFF circuit was presented to show the utility of cell library designed. In Sect. 10.5, the verification of the proposed filter is explained. In Sect. 10.6, simulation results and comparison with CMOS based FIR is provided. Finally, remarks and conclusion are proposed in Sect. 10.7.

10.2 Design Approach and Device Modelling 10.2.1 Memristor Ratioed Logic Design Memristor Ratioed Logic MRL design, is based on the integration of CMOS technology with Memristor to implement varieties of logic gates. Pure Memristive design can be used to design AND/OR gates, while NAND and NOR can be designed by connecting the output of AND/OR Memristive gates to CMOS inverter. This arrangement is not only to obtain NAND and NOR gates but also overcomes the signal degradation problem. MRL method is a voltage-based model, hence the logical state of MRL is defined by the output voltage level, where, voltage level represents the high state “1” and the low state “0” by the high voltage and the low voltage respectively.

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Table 10.1 RRAM memristor parameters for simulation Parameter

I e (A)

G0 (m)

L (nm)

Gmax (nm)

Gmin (nm)

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6.14e−5

2.75e−10

5

6e−12

3.14e−14

10.2.2 Device Modeling Metal oxide-based resistive switching memories RRAMs are intended for use in wide range application of nonvolatile memory. In this paper, the Memristor based RRAM model [11], has been employed to implement the Memristive behavior of the devices in which resistance varies depending on the value, direction, and duration of the applied voltage. The device resistance is altering between low resistance state (LRS) and the high resistance state (HRS). The current is dependent on the oxide layer state while the rate of vacancy generation (E ag )/recombination (E ar ) has a direct impact on the oxide layer state. A Memristor based RRAM is simulated in Cadence Virtuoso. The utilized parameters for the model are specified in Table 10.1. Where I e , G, G0 , are hopping current density, gap length which can either minimum value Gmin or maximum value Gmax based on the applied voltage and window resistance coefficient.

10.3 Finite Impulse Response Filter Design Finite Impulse Response FIR filter is a core element in most applications of signal and image processing [12]. In these applications, low power, area efficient, and high speed are required when designing these filters. Several stages are involved in the design and implementation of FIR filters based on the number of required coefficients. In this paper, a low pass FIR filter with 15 coefficients is considered. Therefore, the designed filter has 15 stages. The characteristic of this filter is shown in (10.1) Y [n] = b0 X [n] + b1 X [n − 1] + · · · bk X [n − k]

(10.1)

Several procedures are involved in the implementation of the proposed filter as shown in Fig. 10.1. The design starts by describing the specification of the filter using Verilog High Definition Language (HDL). The hardware description should be successfully verified and tested by NC-Verilog Cadence simulator using a welldesigned test bench. Next, Synopsys design compiler goes through the descriptive Register-Transfer Level (RTL) in order to convert it to CMOS based gate netlist. This conversion is achieved from the characterized gates data provided by the CMOS standard cell library. Consequently, the gate netlist is verified and tested with the same test bench. Besides, the generated CMOS gate netlist is extracted to obtain all gates involved in the design. Finally, the Memristive gates which have been implemented and tested using MRL method were characterized to build standard Memristive cell

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(a) netlist CMOS Base

(b) netlist memristive Based,

(C) design layout

Fig. 10.1 Flow chart displaying design flow based on Synopsys EDA tool for proposed FIR filter

library to be utilized by Synopsys compiler to generate the Memristive netlist gates. The synthesized FIR memristor based netlist gates should be verified and tested again with the test bench.

10.4 Case Study The core of this work is to create a standard cell library for Memristive gates based MRL design. This procedure is very sensitive since requires implementing and validating all gates in the design. As an example, a sequential D-Flip Flop (DFF) circuit shown in Fig. 10.2. has been chosen to describe all steps involved in the design procedure. DFF has been implemented in the Cadence schematic level, verified in the behavioral level with NC-Verilog, and characterized its Memristive logic gates by Cadence tools. All characterized gates data is fed to the standard library. This library contains

Fig. 10.2 a D-latch schematic circuit. b MRL based D-latch. c Implemented memristor based DFF

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information about input capacitance of each pin of DFF, output net capacitance, transition time, values of rise/fall delay and power consumption. The synthesis tools rely on capacitance values to compute delay and dynamic power. Therefore, it is vital to calculate the capacitance values, for each pin “input/output” where, C is calculated based on (10.2) i C= VDD

t2 i(t)dt

(10.2)

t1

Here i(t) is the current passing through DFF. Since capacitance values are determined, the synthesis compiler can carry out the power consumption based on the energy values provided by the library. All energy values are kept in the library table after measuring DFF consumed energy. Equation (10.3) is used to calculate dynamic power consumption of the designed circuit. PD = αC f VDD

(10.3)

Here α, C, and f, are switching activity factor, capacitance and design frequency respectively. In addition, the DFF propagation delay is computed by measuring the time interval between the input slew and output slew. The slew for DFF alteration times is defined as the time when signal rise from 30% to the 70% and fall from 70 to 30% of its VDD. While DFF area approximation is linked directly to the size of CMOS inverter in every cell. Designed Memristive DFF based MRL has 16 memristors and 14 MOSFET. Therefore, memristors are loaded on inverters between the upper layer of CMOS metal. Figure 10.3 shows both MRL based D-latch and DFF simulation results in the schematic level.

Fig. 10.3 Simulation results in schematic level a MRL-based D-Latch. b MRL-based DFF

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10.5 Design Verification The verification of the proposed filter is achieved by making a comparison between the Memristive based filter output signals and the expected output signal which has been generated in MATLAB by the designed Verilog test bench. This mechanism of verification can be illustrated as shown in Fig. 10.4.

Fig. 10.4 a Organization of the proposed FIR. b Verification flowchart for proposed FIR

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It involves primarily several modules which are: , , and . and are responsible for loading input signals “8 bits every cycle” and write the filtered signals in the output file “hex decimal format” respectively. Whereas, is responsible for storing data temporarily and sending out undesired frequencies. And is an address generator uses flags to make all the design modules synchronized under one main clock “CLK” which is lead to a great impact on speed enhancement and power consumption. Thus, signals flow throughout the modules are explained as flow. input signals received and stored temporarily in register, and then signals are sent serially every clock cycle to where signals are stored again in another register, also communicates with in order to use the generated addresses to find the desired signals and pass it serially to as a final stage. At the end, the desired signals are written in the form of the array corresponding to the filtered signals.

10.6 Experimental Results Designing the hybrid Memristor-CMOS based FIR has been executed through different stages, from implementing the filter in the schematic level to test and verify the filter behavior in Cadence “TSMC 180 nm cell library”. The design utilized a Memristor based RRAM device. All related parameters for the device provided in Table 10.1, were employed to implement the Memristive behavior of the device. The filtering process of the proposed filter was tested by applying a set of signals with different frequencies. The simulation results have shown the behavioral model of the design was capable of distinguishing between all input signals and passes only signals with the desired frequency as shown in Fig. 10.5. One significant aspect of this proposed architecture was the design of the standard cells Memristive library which contains all characterized cells employed in the design

Fig. 10.5 a Input signals in frequency domain at 600 and 2000 Hz. b Filter output signal in frequency domain at 600 Hz

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Table 10.2 Characterized cells employed in the design Gate

Description

Equation

AND2X1

Provides the logic AND for two inputs

Y = (A · B)

OR2X1

Provides the logic OR for two inputs

Y=A+B

XOR2X1

Provides the logic Exclusive OR for three inputs

Y=A⊕B⊕C

INVX

Provide logical inversion of single input

Y=A

ADDFX2

Provide arithmetic sum S and carry out C

S = (A ⊕ B ⊕ C, C = (A ⊕ B) · C + (A · B)

DFFHQX1

positive edge triggered static D type flip flop

Case study

NOR2X1

Provide logical NOR

Y = (A + B)

NAND2X1

provides the logical NAND of two inputs

Y = (A · B)

are displayed in Table 10.2, such cells are OR, NOR, AND, NAND, DFF and variety of other cells with specified Boolean function. In this paper, a case study of the DFF circuit was presented to show the accomplishment of cells characterization process. Synopses synthesis tool utilizes the built library based memristor to deliver the DFF netlist and provide measurements such as power consumption, area, and delay as shown in Table 10.3. The results have proven that Memristive DFF with 16 memristors and 14 MOSFET has less area comparing to DFF based CMOS. The design memristor base FIR appears to be compact comparing to CMOS base FIR. Consumed power and delay are comparably reduced as well, as shown in Table 10.4. Table 10.3 Results DFF memristor based results

Table 10.4 Comparison of experimental results

Design

Area (µm−2 )

Power (µW)

Delay (ns)

CMOS

459.04

83.60

0.37

Memristor

379.20

41.29

0.16

Design

Area (µm−2 )

Power (mW)

Delay (ns)

CMOS

95813.62787

3.3099

0.36

Memristor

10973.69952

0.8776

0.31

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10.7 Conclusion In this paper, a hybrid memristor-CMOS based FIR filter design has been proposed with Memristive RRAM devices. The proposed filter has been verified and tested at different stages of the design in Cadence environment, NC-Verilog and Matlab. The simulation results have indicated that the filter was effectively able to distinguish between all inputs signals and allow only the desired signals to pass through the passband region and reject any signal beyond the stopband frequency.

References 1. H.A.D. Nguyen, L. Xie, M. Taouil, S. Hamdioui, K. Bertels, Synthesizing hdl to memristor technology: a generic framework, in 2016 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH), July 2016, pp. 43–48 (J. Latex Cl. Files 11(4), December 2012 5) 2. S. Shin, K. Kim, S. Kang, Memristor applications for programmable analog ics. IEEE Trans. Nanotechnol. 10(2), 266–274 (2011) 3. Y.V. Pershin, M. Di Ventra, Practical approach to programmable analog circuits with memristors. IEEE Trans. Circuits Syst. I Regul. Pap. 57(8), 1857–1864 (2010) 4. M. Teimoory, A. Amirsoleimani, J. Shamsi, A. Ahmadi, S. Alirezaee, M. Ahmadi, Optimized implementation of memristor-based full adder by material implication logic, in 2014 21st IEEE International Conference on Electronics, Circuits and Systems (ICECS), Dec 2014, pp. 562–565 5. F.S. Marranghello, V. Callegaro, A.I. Reis, R.P. Ribas, Four-level forms for memristive material implication logic. IEEE Trans. Very Large Scale Integr. (VLSI) Syst. 1–5 (2019) 6. S. Kvatinsky, D. Belousov, S. Liman, G. Satat, N. Wald, E.G. Friedman, A. Kolodny, U.C. Weiser, Magicmemristor-aided logic. IEEE Trans. Circuits Syst. II Express Briefs 61(11), 895–899 (2014) 7. S. Kvatinsky, N. Wald, G. Satat, A. Kolodny, U.C. Weiser, E.G. Friedman, Mrl memristor ratioed logic, in 2012 13th International Workshop on Cellular Nanoscale Networks and their Applications, Aug 2012, pp. 1–6 8. V. Ntinas, I. Vourkas, G.C. Sirakoulis, LC filters with enhanced memristive damping, in 2015 IEEE International Symposium on Circuits and Systems (ISCAS), May 2015, pp. 2664–2667 9. H.A.D. Nguyen, L. Xie, M. Taouil, S. Hamdioui, K. Bertels, Synthesizing hdl to memristor technology: a generic framework, in 2016 IEEE/ACM International Symposium on Nanoscale Architectures (NANOARCH), July 2016, pp. 43–48 10. F.M. Bayat, F. Alibart, L. Gao, D. Strukov, A reconfigurable fir filter with memristor-based weights. arXiv preprint arXiv:1608.05445 (2016) 11. P. Chen, S. Yu, Compact modeling of rram devices and its applications in 1t1r and 1s1r array design. IEEE Trans. Electron Devices 62(12), 4022–4028 (2015) 12. R.C. Gonzalez, R.E. Woods, Digital Image Processing (Prentice Hall, New Jersey, 2002)

Chapter 11

Chaotic Oscillator for LPWAN Communication System A. Litvinenko, A. Aboltins, D. Pikulins and F. Capligins

Abstract The paper is devoted to the study of mathematical model of chaotic oscillator, as well the ability of its employment in low-power wide-area network (LPWAN) communication systems. For this reason, modified Chua circuit with four state variables and nonlinear function is presented and analyzed. The impact of system’s parameters on the dynamics of the oscillator is investigated and three basic modes of operation are found. The obtained chaotic modes of the proposed dynamical system could provide possibility to build secure and robust communication system, moreover the variations of system parameters could be used for implementation of multi-user communication system. Besides that, the chaotic synchronization method, based on the drive-response system employment with linear feedback is described and its properties are studied for differing initial conditions of state variables in drive and response oscillators. The most significant results of research are presented in the conclusions part.

11.1 Introduction In recent years, wireless communications technology has found a growing range of applications in our daily lives. One of the most significant technological breakthroughs is the development and integration of wireless sensor networks (WSNs). With the development of the Internet of things (IoT) concept, the importance of the role of WSN for automated data collection is growing rapidly. Therefore, the capabilities and technologies of WSN are currently at an advanced stage of development—research is being conducted to improve existing techniques and introduce the new ones. This paper studies the applicability of the certain chaotic oscillator to the development of wireless digital communication system, that could provide a possible alternative to the existing low-power wide-area network (LPWAN) systems. The use A. Litvinenko · A. Aboltins · D. Pikulins (B) · F. Capligins Institute of Radioelectronics, Riga Technical University, Riga, Latvia e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_11

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of chaotic signals in telecommunications is not a new technique and has its own advantages: it is very difficult to intercept a chaotic signal, it has a good resistance to multipath and narrowband interference, as well as the ability of chaotic synchronization [1]. Chaos-based communication systems have not yet received wide recognition among other LPWAN and WSN, therefore the field of application of this technology is being studied. Since the proposed communication system will be based on the chaotic oscillator and the properties of chaotic signal can be explored, the influence of the parameters of the oscillator on its dynamics will be investigated and the synchronization properties will be studied. The main objective of the article is the analysis of the performance of the chaotic oscillator, as well as evaluation of the ability to build a communication system based on it. The study is based on the simulations of the chaotic oscillator within MATLAB/Simulink simulation tools. The paper consists of four sections: the first section represents introduction, the second section discusses the mathematical model of the chaotic oscillator, the third section presents the impact of the dynamics on the system parameters, the fourth section describes the chaotic synchronization, and the conclusions summarize the results and set further research steps.

11.2 Mathematical Model of Chaotic Oscillator Nonlinear dynamical systems that demonstrate complex chaotic behavior at certain parameters and initial conditions are called chaos generators. Such systems can be described by a mathematical model using equations of state variables and those systems can be implemented physically as mechanical, electronic, or other devices. Following features of Chaotic systems [1] make them attractive to the communications: • low energy consumption—circuits that generate chaotic signals can work efficiently with very little power; • noise-like waveform—the chaotic system output signal is aperiodic and by nature is noise-alike, what makes it difficult to detect a chaotic signal; • Broad spectrum—the chaotic system’s autocorrelation function decreases very rapidly because power spectrum of chaotic signal is broad. In addition, the correlation between chaotic signals from the same system but with different initial conditions is very low due to the system’s extreme dependence on the initial conditions; these features make chaotic generators very suitable for spread spectrum communications; • Nonlinearity—chaotic systems have a nonlinear nature that is useful for communication systems because of the ability to increase spectral efficiency by encoding more information on the frequency band than using linear system; • Synchronization—ability of one chaotic system synchronize its state with the state of another system can be efficiently used in various communication applications.

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Fig. 11.1 a Circuit diagram of Chua circuit; b I-V curve of Chua diode (N R )

As a clear example of a chaotic generator, the Chua circuit [2–5] is considered to be one of the simplest dynamical systems that has both a physical and a mathematical model. It was described in 1983 by Professor Leon Chua from the University of California. As can be seen from Fig. 11.1a, Chua circuit consists of two capacitors, one coil, one resistor and a nonlinear element called Chua diode which in fact is an active element with negative resistance (see Fig. 11.1b). Using Kirchhoff and Ohm laws it is possible to obtain state variable equations: C1 dudt1 = R1 (u 2 − u 1 ) − f (u 1 ) C2 dudt2 = R1 (u 1 − u 2 ) + i 3 L didt3 = −u 2 ,

(11.1)

where C 1 , C 2 —capacities of first and second capacitor, R—resistance, L—inductance of coil, u1 —voltage on the first capacitor with capacitance C 1 , u2 —voltage on the second capacitor with capacitance C 2 , i3 —current through coil. Nonlinear I-V curve which characterizes the Chua diode is described by following equation: 1 f (u 1 ) = i N R = G b u 1 + (G a − G b )(|u 1 + E| − |u 1 − E|) 2

(11.2)

where i N R —current through Chua diode, Ga and Gb —steepness of linear segment. Points with u1 = E, and u1 = −E—correspond to the points where the characteristic curve changes the steepness. By replacing variables in the system of (11.1) by dimensionless variables x, y, z we obtain:

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= α(y − x − f (x)) dy =x−y+z dτ dz = βy dτ

dx dτ

(11.3) And nonlinear I-V curve now becomes: 1 f (x) = bx + (a − b)(|x + 1| − |x − 1|) 2

(11.4)

In turn, dimensionless variables are defined as: y = uE2 , z = i 3 ER 2 α = CC21 , β = R LC2 a = RG a , b = RG b , τ = |RCt 2 | . x=

u1 , E

(11.5)

In this research a modified Chua circuit—chaotic generator with four state variables is used. This modification of the chaotic system allows to create more diverse chaotic generators, which is important feature for enabling simultaneous transmission in the multi-user communication system. Increase of number of parameters leads to growth of number of possible combinations when oscillator works in the chaotic mode. MATLAB Simulink model of two modified Chua circuit-based generators, which can be synchronized with each other, available from [6] has been used. The modified Chua-circuit based generator can be described by following system of equations: d p1 dt d p3 dt

= −g( p1 − p3 ) − p2 d p2 = p1 + γ p2 dt = θ (g( p1 − p3 ) − p4 ) d p4 = σ p3 . dt

(11.6)

where p1 , p2 , p3 , p4 are state variables, and g(p1 − p3 ) is nonlinear function:  g( p1 , p3 ) =

c( p1 − p3 − d) ( p1 − p3 ) > 1 0 ( p1 − p3 ) ≤ 1

(11.7)

Variables θ , σ , γ , c, d are real coefficients of the differential equation, which in the remaining part of the paper are called system parameters. Taking into account that number of state variables of the (11.6) is four, for the visualization of the phase portrait of the chaotic system six independent planes are displayed in Fig. 11.2, for employed initial conditions and coefficients presented in Table 11.1.

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Fig. 11.2 Projections of the attractor of modified Chua circuit (11.6) to different phase planes Table 11.1 Initial conditions and parameters of modified Chua circuit employed in the simulation Initial conditions

Parameters

p1

p2

p3

p4

γ

θ

σ

c

d

0.05

0.06

0.07

0.08

0.5

10

1.5

3

1

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Table 11.2 Weights of the state variables employed in the simulations

k1

k2

k3

k4

−2.6302

−0.6054

0.5870

0.7763

The chaotic signal R is made from the algebraic sum of the state variables pi , which are multiplied by corresponding weights ki , see Table 12.2. In order to create conditions for the synchronization, the sum includes also value of the nonlinear function g( p1 , p3 ): R = p1 k1 + p2 k2 + p3 k3 + p4 k4 + g( p1 , p3 ).

(11.8)

11.3 The Influence of System Parameters on the Dynamics of the System The mathematical model of the Chua circuit under investigation includes the following coefficients: γ , θ , σ , c and d. Coefficients c and d define the influence of nonlinear function (11.7), however it should be noted, that the mentioned function is present only in the drive generator, whereas in the response system the same function is obtained through the linear feedback. As a result, it has been decided not to change the values of coefficients c and d, but study the dependence of the dynamics of the system on different values of coefficients γ , θ and σ . 64 different combinations of coefficients γ , θ and σ values have been tested. The results have been interpreted on the basis of the projection of the attractor of the generator in the p1 p2 plane, as well as waveforms of the generated signal R(t). Time of simulation is 1000 s, the initial conditions are given in Table 11.1, while weights of the state variables in Table 11.2. The simulation uses the Simulink ode45 (Dormand-Prince) solver with relative precision 10−10 and absolute precision 10−16 and dynamic solver step. Simulation results are depicted in the Table 11.3. The numeration of parameter combinations is obtained for corresponding sequential increase of γ , θ and σ values. During the simulations different areas with chaotic dynamics have been obtained, whereas the other values lead to the development of periodic oscillations or the exponential divergence to infinity. The corresponding states, that are shown in Table 11.3 correspond to: 1. Chaotic mode. Attractor in the observed phase plane is projected as dense trajectory orbits, that differ from the projection of attractor for with the initial system parameters, but the relative difference is not too obvious. These modes of operation could be efficiently exploited for the development of chaos-generator based LPWAN communication system, so the corresponding parameter values are highlighted in the table. The example of waveform of the chaotic signal could be observed in the Fig. 11.3.

−0.5

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.46

0.5

0.5

0.5

0.5

0.5

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

σ

3

3

2 1.2

0.6

1.5

1.5

1.5

1

3.2

−10

2

1.5

2

1.4

2

1.4

1.4

1.4

1.4

1.4

1.4

1.3

2

1.4

1.5

1.1

18

10

5

3

1

3

2

1.5

1.4

1.2

1.1

1.1

18

3

θ

10

γ

Nr.

3

3

3

3

3

1

2

1

1

1

1

1

1

1

1

1

1

3

2

2

2

mode

53.

52.

51.

50.

49.

48.

47.

46.

45.

44.

43.

42.

41.

40.

39.

38.

37.

36.

35.

34.

33.

Nr.

Table 11.3. The dependence of the dynamics of the system under study on parameters γ, θ and σ γ

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.6

0.5

0.5

θ

18

15

12

10.5

10.5

10.1

8

8

8

8

8

8

8

8

7

5

3

3

3

10

10

σ

2

1.7

1.7

1.7

1.6

1.6

5

3

2.5

2

1.8

1.7

1.3

1.1

1.7

1.7

1.6

1.5

1.4

1.5

1.4

1

1

1

1

1

1

3

3

3

1

1

1

3

3

3

3

3

3

3

1

(continued)

mode 1

11 Chaotic Oscillator for LPWAN Communication System 107

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

γ

22.

Nr.

Table 11.3. (continued)

θ

10

10

9

9

9

8

5

4

3

3

3

σ mode

3

3

−1.5 1

1

1

3

1

1

1

1

1

3

1.5

1.4

0.5

1.5

1.5

1.5

1.5

1.4

1.3

64.

63.

62.

61.

60.

59.

58.

57.

56.

55.

54.

Nr.

1

0.8

0.8

0.7

0.7

0.7

0.7

0.6

0.6

0.6

0.6

γ

θ

10

18

8

18

10.5

9

3

100

36

30

20

σ

1.5

2

1.8

2

1.7

1.6

1.7

1.7

2

1.7

1.7

mode

3

3

3

3

3

3

3

1

1

1

1

108 A. Litvinenko et al.

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Fig. 11.3 Chaotic oscillations of modified Chua system for parameter set Nr.30 (see Table 11.3)

2. Periodic oscillations. The projection of attractor includes a single fixed orbit. The example of the waveform is shown in the Fig. 11.4. 3. State variables exponentially diverge to infinity. The example of the mentioned divergence is depicted in the waveform in the Fig. 11.5.

Fig. 11.4 Periodic oscillations of modified Chua system for parameter set Nr.1 (see Table 11.3)

Fig. 11.5 Divergence to infinity of modified Chua system for parameter set Nr.20 (see Table 11.3)

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During the following investigation, the parameters of the generator were left fixed. Their values correspond to the ones defined in the Table 11.1 that revealed to be quite reliable solution for the development of the communication model.

11.4 Analysis of Chaotic Synchronization The synchronization of chaos generators is explained in many sources, some most noticeable are [1, 7–15]. From the definition of chaos, it follows that the difference between two independent and identical chaotic systems with similar initial conditions increases exponentially. However, in a linked system consisting of two identical chaos generators with different initial conditions there is an ability of self-synchronization. The synchronization of two identical chaos generators, respectively described by n-dimensional state variables x(t) and y(t), is present if the difference |x(t) − y(t)| converges to zero when the time approaches infinity. As a result of synchronization, the state variables of the two chaos generators are identical, even though the generators started their operation from completely different starting conditions. The stability of can be synchronization between two dynamical systems is classically assessed by the Lyapunov master stability function [14, 16], which includes the estimation of the difference of the trajectories of the respective states of the two systems. During the practical evaluation of chaotic generator synchronization, it is not always possible to use the Lyapunov function, as there must be direct access to the state variables of the two systems. Therefore, the synchronization of two (leading and controlled) chaos generators evaluated in the following two ways: 1. Evaluation of the correlation between the output sequences of the two generators. If synchronization occurs, the correlation coefficient is approaching one, as time approaches infinity. 2. Estimation of the root mean square (RMS) of the error signal obtained from the difference between the output sequences of both systems. When synchronization occurs, the RMS of the error signal approaches zero when time approaches infinity. To make synchronization possible, systems need to be linked in a certain way. This research uses a synchronization method based on linear feedback as described in [1, 7]. In this method, which employs two identical systems—drive system and response system, synchronization is achieved by linear feedback. The block diagram is depicted in Fig. 11.6. Fig. 11.6 Block diagram of linear feedback-based synchronization

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Fig. 11.7 Block diagram of synchronization method based used in the research

Two identical systems x(t) and y(t) produce outputs x  (t) and y (t) determined by their state variables. The output could be just one state variable, partial or full sum of variables, etc., but calculated identically in both systems. Signal of the drive system is processed and sent over the link to the receiving side. After reception of the signal from the drive system, the output of the response system is subtracted from the received one, produced by the drive system, generating an error signal e(t). This signal is further used to adjust the operation of the response system until the error signal approaches zero, which means that the systems become synchronized. One of synchronization methods based on the linear feedback can be used to synchronize chaotic systems based on the modified Chua circuit. The drive system shown in Fig. 11.7 is based on (11.6) and (11.7) and the transmitted chaotic signal R(t) (11.8) (Fig. 11.8) is based on the sum of the weighted state variables and nonlinear function. The output sequence R of the chaos generator is non-periodic with emphasis on low frequency components. The receiver contains four state variable functions p 1 , p 2 , p 3 , p 4 , however, it lacks nonlinear function g( p1 , p3 ), which is derived from the transmitted signal R using the linear feedback (see Figs. 11.7 and 11.8). The waveform of the nonlinear function g( p1 , p3 ) can be viewed in Fig. 11.9 and it contains only positive values or zero. This feature can be used to create a digital communication system which uses chaotic self-synchronization property for data transmission and for synchronization state estimation uses the values of a recovered nonlinear function. At the first moment the chaotic sequence R(t) in the response system is perceived as the nonlinear function g( p1 , p3 ). The sum of the variables pi is fed back to the input of receiving system, multiplied by the same respective weight coefficients. If the initial conditions for both systems are the same, synchronization occurs immediately. At different initial conditions, the synchronization occurs quite quickly, as it is demonstrated by the simulation results, shown in Fig. 11.10 for the case when, initial conditions of drive and response systems differ 10 times—see Table 11.4. Figure 11.10 shows the logarithm of magnitude (absolute value) of the difference between values of respective state variables in drive and response chaos generators. From Fig. 11.10 it can be concluded that error decreases exponentially

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Fig. 11.8 Waveform of the generated chaotic sequence R

Fig. 11.9 Waveforms of the signal of nonlinear function g(p1 , p3 )

and in the current model the synchronization can be considered as established in approximately 26 s, as the difference between system state variables takes values around 10−16 , which is the numerical rounding threshold of the software. Also, it can be seen that the dynamics of the error decrement for each state variable is different and it is related to the weight coefficients as well as the differences in the initial conditions. Speed of the synchronization determines duration of the symbols to be transmitted and affects data rate of digital communication system. If we want to achieve high immunity to the interference, symbol duration in the communication system should be greater than 26 s, whereas keeping of signal bandwidth and reduction of symbol interval below 26 s will lead to increase of the data rate cost of reduced immunity to the noise and synchronization issues. It is worth to mention that data rate of the communication system can be adjusted also by

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Fig. 11.10 Assessment of the synchronization speed using measurement of magnitude logarithm of difference between values of state variables

Table 11.4 Parameters used for the simulation of the synchronization Initial conditions of the drive system

Initial conditions of the response system

p1

p2

p3

p4

p1

p2

p3

p4

0.05

0.06

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changing of time scale of the dynamical system. This can be done either by adjusting the coefficients of the differential equations or, in case of discretized dynamical system—sample rate of the model. However, in this case the bandwidth of the data link will change as well.

11.5 Conclusions The chaotic oscillator employment for LPWAN communication system could increase its robustness, security, noise immunity, and could be used for synchronization due to the inherent properties of chaotic systems. In this research chaotic oscillator based on a modified Chua circuit has been studied via simulation in MATLAB Simulink. The introduced chaotic oscillator was defined as the mathematical model with four state variables and a nonlinear function. The impact of the dynamics of the system under study on 64 parameter combinations was investigated, revealing 33 combinations corresponding to chaotic behavior, 4 to periodic oscillations and

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27 to the exponential divergence to infinity. Only the obtained chaotic modes are applicable to the development of LPWAN communication system. The differences in the system parameters of chaotic oscillators could be used for implementation of multiple access functionality. Moreover, the chaotic synchronization of drive-response dynamical systems with linear feedback was presented and the required synchronization time was studied for one of the observed chaotic modes. Since four state variables functions are similar for both drive and response systems, the nonlinear function is transmitted to the response system together with the sum of all state variables. The experiments on chaotic synchronization of two systems with various initial conditions have shown, that, the difference between corresponding state variables was less than 10−15 after system synchronization, providing the ability to use chaotic synchronization for the development of communication system. The shape of the nonlinear function gives the opportunity to use its values for chaotic synchronization identification in the receiving system, simplifying the possible LPWAN communication system design. Next research steps could include the study of noise immunity of the chaotic synchronization, the influence of the initial conditions on the required synchronization time, as well as the development of the concept of LPWAN communication system based on the proposed chaotic synchronization. Acknowledgements This work has been supported by the European Regional Development Fund within the Activity 1.1.1.2 “Post-doctoral Research Aid” of the Specific Aid Objective 1.1.1 “To increase the research and innovative capacity of scientific institutions of Latvia and the ability to attract external financing, investing in human resources and infrastructure” of the Operational Programme “Growth and Employment” (No.1.1.1.2/VIAA/2/18/345).

References 1. P. Stavroulakis, Chaos Applications in Telecommunications (CRC Press, Taylor & Francis Group, 472 pp., 2006) 2. R. Lozi, Designing chaotic mathematical circuits for solving practical problems. Int. J. Autom. Comput. 11(6), 588–597 (2014) 3. M. Chen, J. Yu, B.-C. Bao, Hidden dynamics and multi-stability in an improved third-order Chua’s circuit. J. Eng. (10), 322–324 (2015) 4. S.M.S. Kang, Singularities of nonlinear circuit theory and applications: achievements of Professor Leon Ong Chua. IEEE Circuits Syst. Mag. 18(2), 10–13 (2018) 5. M.Z. De la Hoz, L. Acho, Y. Vidal, A modified Chua chaotic oscillator and its application to secure communications. Appl. Math. Comput. 247, 712–722 (2014) 6. T. Hoang, Simulink model for observer based synchronization in Chua’s systems [viewed 12 February 2019]. https://www.mathworks.com/matlabcentral/fileexchange/26246-observerbased-synchronization-in-chua-s-systems 7. M. Hasler, Synchronization principles and applications, in IEEE International Symposium on Circuits and Systems ISCAS ’94, pp. 314–327 (1994) 8. J. Zhang, L. Zhang, X. An, H. Luo, K.E. Yao, Adaptive coupled synchronization among three coupled chaos systems and its application to secure communications. Eurasip J. Wirel. Commun. Netw. 1–15 (2016)

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9. G. Kaddoum, D. Roviras, P. Chargé, D. Fournier-Prunaret, Robust synchronization for asynchronous multi-user chaos-based DS-CDMA. Sig. Process. 89(5), 807–818 (2009) 10. Ö. Morgül, E. Solak, Observer based synchronization of chaotic systems. Phys. Rev. E 54(5), 4803–4811 (1996) 11. L.M. Pecora, T.L. Carroll, Synchronization of chaotic systems. Underst. Complex Syst. 48, 101–133 (2009) 12. A. Senouci, A. Boukabou, K. Busawon, A. Bouridane, A. Ouslimani, Robust chaotic communication based on indirect coupling synchronization. Circuits, Syst. Signal Process. 34(2), 393–418 (2015) 13. X. Wu, Y. He, W. Yu, B. Yin, A new chaotic attractor and its synchronization implementation. Circuits, Syst. Signal Process. 34(6), 1747–1768 (2015) 14. F. Wu, J. Ma, G. Ren, Synchronization stability between initial-dependent oscillators with periodical and chaotic oscillation. J. Zhejiang Univ. A 19(12), 889–903 (2018) 15. D. Yang, Networked reliable synchronization of chaotic Lur’e systems with sector- and Sloperestricted nonlinearities. Circuits, Syst. Signal Process. 35(5), 1665–1675 (2016) 16. L.M. Pecora, T.L. Carroll, Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80(10), 2109–2112 (1998)

Chapter 12

Effects of Control Non-idealities on the Nonlinear Dynamics of Switching DC-DC Converters D. Pikulins, S. Tjukovs and J. Eidaks

Abstract Recent researches have shown that classical models used for the analysis of the dynamics of switching power converters do not allow the prediction of wide variety of nonlinear regimes, endangering the operation of power systems and all the connected circuitry. One of the causes is that the majority of models and corresponding stability criteria are based on the assumption, that the control elements with momentary reaction are used. The paper is dedicated to the investigation of the effects of control system’s non-idealities on the global dynamics and stability of switching power converters. The discrete-time model of boost converter, including switching delays, is presented and studied by means of innovative approach, based on the numerical calculation of periodic regimes, with the following definition of stability regions in parameter plane. It is shown that the time lag of feedback loop substantially changes the topology of stability borders as well as leads to sudden qualitative changes in the dynamics of the system. The obtained results allow the identification of critical control parameters’ values and their tolerances, excluding the possibilities of catastrophic phenomena in the dynamics of switching power converters.

12.1 Introduction The ever growing demand for portable power supplies leads to the increase of operating frequencies of most widely used types of DC-DC conversion devices—switching power converters (SPC), allowing the use of much smaller power plant components, consequently diminishing the required printed circuit board area. Thus, the switching frequencies (f sw ) of SPC during last decade have grown from tens of kilohertz to several megahertz [1], leading to the noticeable miniaturization of these devices. On the other hand, it appeared that some of the unavoidable non-idealities of the control circuitry, that could be ignored while operating at relatively low frequencies, D. Pikulins (B) · S. Tjukovs · J. Eidaks Institute of Radioelectronics, Riga Technical University, Riga, Latvia e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_12

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play an important role in the formation of dynamics and stability of SPC at higher frequencies. One of the most noticeable negative effects to be taken into account is the delay in the components of the analog control loop and the inherent delays of switching elements—such as transistors and corresponding drivers. The values of delays may reach up to 10–20% of the whole switching period [2], causing the unpredictable stability issues in the operation of SPC. The situation becomes even worse as the digital implementation of the control circuitry is becoming more popular. Typical delay values in the feedback systems including digital signal processors and analogto-digital converters are 30–100% of switching period [3]. The vast majority of previous researches have been devoted to the investigation of nonlinear dynamics of SPC at relatively low frequencies, assuming momentary reaction of switching circuitry. The first article, studying the effects of non-idealities on the switching actions in SPC and defining the appearance of new dynamical sequences—“missed switchings”—was published in 2004 [4]. In this paper the author provides some preliminary results and demonstrates that feedback delays and switching spikes could drastically change the structure of bifurcation patterns and cause the appearance of sudden loss of stability of main period-1 (P1) regime, as well as qualitative changes in the chaotization scenarios of buck converter under current mode control (CMC). The analysis is based on simple 1D map, assuming that the optimal output filter is used, that is not the case in many practical applications, especially taking into account tolerances and degradation of electrolytic capacitors used in the mentioned filters [5]. The author provides some kind of experimental verification of results and highlights that for more accurate and deep investigation of the observed phenomena the discrete-time models of higher order are required. Another group of authors has investigated the dynamics of the same switching electrical circuit by means of more complete 2D map [6], using the same circuit parameters and magnitude of the delay as in [4]. Authors focus their attention on the effects of delay on the first bifurcation and reveal that the introduction of delay causes the reduction of the region of stable P1 operating regime as well as leads to the appearance of coexisting attractors. Results presented in [7] could be thought as logical continuation of [6]. The authors provide the iterative map, defining not the actual dynamics of SPC, but the model, whose action is “the same as current mode controlled SPC” and using the same delays as in [4, 6]. The main conclusions, that could be made from [7] are that the introduction of delay effects the period doubling (PD) as well as border collision (BC) bifurcations, defines new structure of return map and new types of bifurcations (that could not be observed in systems with ideal switchings). Is should be noted, that the switching period T, chosen as the secondary bifurcation parameter within this investigation, is commonly selected at the very beginning of the design process of SPC, defining the most general parameters of the converter (such as characteristics of the output filter, switching elements and the control circuitry). Thus, in classical control schemes T should not generally change during the normal operation of SPC. So the most obvious candidates to the secondary bifurcation parameter are inductance L, output capacitance C (the values of which may degrade with time) as well as output

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load RL , input voltage V in and reference current I ref that are commonly varied during the operation of SPC. The results presented in some previous papers are dedicated to the investigation of current mode control in buck converter. However, the greatest advantages the current mode control loop exhibits when applied to the boost or buck-boost converters, where the so called problem of right-half plane (RHP) zero is of great importance [8] and could not be eliminated by means of simple voltage mode control (VMC). Taking into account the achievements of previous authors and the mentioned shortcomings of their investigations, this paper is dedicated to the complete bifurcation analysis of current mode controlled boost converter with delays in the control circuitry. First, the discrete-time model of boost converter is presented, taking into account the possibility to change output capacitance, and introducing the delayed switching action. The proposed model is used for the complete bifurcation analysis on the basis of one and two-dimensional bifurcation diagrams. The analysis is based on the numerical calculation of stable and unstable periodic orbits with consequent continuation as one or several parameters of the system are changed. This allows the detection of all existing periodic regimes and definition of their mutual interconnections as well as explanation of the structure and the nature of appearing chaotic modes of operation. Contrary to commonly used brute-force approach, the mentioned technique allows the detection of coexisting periodic modes, small regions of stable regimes (found during the continuation of unstable branches of bifurcation diagrams), as well as the investigation of effects of collisions with switching surfaces on the structure of stable and unstable modes of operation, that generally define the structure of chaos. The rest of the paper is organized as follows. In Sect. 12.2 the discrete-time model of the boost SPC with non-ideal switchings is presented. Section 12.3 provides the results of complete bifurcation analysis on the basis of bifurcation diagrams. Conclusions are provided in Sect. 12.4.

12.2 The Model of the Boost Converter with Delay The object of the investigation, demonstrating the effects of control non-idealities on the global dynamics of the system, is one of the most widely used SPC—boost converter (see Fig. 12.1). The main aim of this circuit is to maintain almost constant output voltage V out greater than the input voltage E, as load resistance RL and input voltage change within definite range. The possibility to obtain V out > E is ensured by the use of energy storage elements—inductor L and capacitor C, as well as the implementation of switching devices—transistor S and diode D, defining the time intervals the energy is accumulated in reactive elements (L and C) and the length of intervals during which the energy is passed to the load. The value of output voltage is regulated by means of definite control loops, providing the comparison of the obtained output voltage (within voltage mode control) or input current (within current mode control) values to some reference values and defining the corresponding control

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Fig. 12.1. The model of the boost converter under current mode control with delay

pulses driving the switch S. The control schemes of boost converters are almost entirely based on the CMC, as the application of this control strategy allows the use of simplified compensation networks, ensuring meanwhile optimal dynamics in all possible modes of operation. Practically current mode control uses both the output voltage V out and the inductor current iL in the process of formation of control signal. Among various implementation schemes of CMC the most widely used is peak current control, employing the peak value of the current flowing through the inductive or the switching element. The main principles of operation of the converter under peak CMC are shown in the Fig. 12.1. The current of the inductor is sensed and compared to some reference value from the voltage feedback loop (formed by R1, R2, COM1 and V ref —see Fig. 12.1), forming the control signal that determines the time instants for the switching element to turn ON or OFF. The most widely used implementation includes the RS flip-flop with CLOCK element connected to the SET input and comparator output to RESET input. Thus the switch S is turned ON when the clock pulse arrives and is turned OFF when the output of the comparator COM2 triggers the R input of the RS flip-flop, that should appear only if the value of iL reaches the I ref . However, the introduction of the delay in the control circuitry leads to some modification in system’s dynamics. As it could be seen from Fig. 12.2b, the time instant at which the switch turns OFF is defined not by the value of reference current I ref , but by the I ref + m1 * Δt d (where m1 = E/L is the slope of the rising inductor current and Δt d is the value of the delay). Thus, the introduction of delay leads to the appearance of two obvious consequences: 1. the value of the peak inductor current (the control of which is of great practical relevance) is no longer defined by the I ref itself and becomes dependent on the introduced delay; 2. new possible dynamical scenarios occur, as it will be shown during the derivation of the model. It should be noted at this point that in general during the study of dynamics of current mode controlled converters the outer voltage loop could be omitted. The application of mentioned simplification without any noticeable consequences on the reliability of results is possible due to the fact that the dynamics of current loop is of order faster than that of voltage feedback loop and the fast scale dynamics

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Fig. 12.2. a–c Possible interactions between two clock instants; d, e two border cases

of the converter becomes almost unaffected by negligible deviations of feedback voltage V fb (see Fig. 12.1). The other essential argument is that in many practical applications, such as voltage conversion from photovoltaic panels, there is no need for the outer voltage loop in the current mode controlled DC-DC converters as the value of reference current itself may be used to perform the maximal power point tracking providing the most efficient operation of the system [9]. The model obtained in this section will not take into account the slow scale dynamics, introduced by the voltage loop (shown in the Fig. 12.1 in grey), focusing instead on subharmonic oscillations of current mode control. The dynamics of this converter could be described by one of the commonly known approaches: • systems of differential equations; • linearized model; • discrete-time model. First and the most obvious way to provide the analysis of the dynamics of SPC is to obtain the systems of differential equations, describing the topology of the converter at definite positions of the switches (S and D). Together with the defined switching surfaces this model offers the most complete description of the converter. However, the simulation of the proposed model would take unacceptable time and computational effort. The second approach, most widely used by practicing engineers, is the application of linearized small-signal models. These models have been developed for the majority of existing SPC [10] and the tools used for the small-signal analysis are well defined and proved to be very useful. On the other hand, it has been shown, that even if the SPC meets the stability criteria of small-signal analysis, the system could become unstable in a variety of ways [11], as these models could not predict the so called “fast scale instabilities” commonly appearing in SPC. The last approach is to obtain the discrete-time model of the converter in order to make the analysis simpler than using differential equations and provide the detailed

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examination of instabilities occurring at the frequencies of the order of f sw . The latter models have one more noticeable advantage—quite straightforward improvement of obtained equations, in case some more detailed analysis of the system under test is needed. In example, it is possible to add some parasitic elements of the power plant (such as resistances of real inductors and capacitors) as well as control circuitry (such as switching spikes and delays). Considering the obvious advantages of the discrete-time modeling, the iterative map of boost type SPC under current mode control including switching delays is obtained. The model consists of three systems of difference equations, describing the evolution of inductor current and output capacitor voltage samples (in and vn accordingly) between two sequent CLOCK pulses, taking into account the possible interactions of the control signals and the inductor current (see Fig. 12.2). The applicability of certain system of equations is defined by the position of the sample of inductor current in with respect to two borderlines, as it is shown later. As it could be seen from Fig. 12.2, three development scenarios are possible: 1. the switch remains ON for the whole period and the operation of the system is governed by equations for ON interval only (see Fig. 12.2a); 2. the inductor current reaches the reference value I ref + m1 * Δt d and the switch turns OFF (if it was in the ON position previously)—see Fig. 12.2b; thus, the dynamics of boost converter is defined by two systems of equations—for ON and OFF intervals; 3. the switch stays OFF all the period (see Fig. 12.2c); this situation is unique to the system with the delay, as then the falling inductor current sample may be situated above the I ref at the time instant the CLOCK pulse arrives; in this case, the switch is not turned ON and the inductor current continues to fall. The transitions between three possible solutions are defined by two borderlines: • the first one (see Fig. 12.2d) defines the border case for the development of the system in the current switching cycle between (1) and (2); • the second one (see Fig. 12.2e) defines the border case for the development of the dynamics of the system in the next cycle between (2) and (3). Schematically all possible transitions and their dependence on the position of the samples of inductor current in relation to borderlines are summarized in Fig. 12.3. The principles of derivation of discrete-time models for simple SPC on the basis of differential equations and inspection of control waveforms is quite straightforward and thoroughly documented (see e.g. [11, 12]), and for the sake of simplicity are omitted here. The final model of the CMC boost converter including the Δt d could be defined as follows. Two borderlines are: Iborder1 = Iref + m 1 (td − T )

(12.1)

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Fig. 12.3. The dependence of the development of the system on the position of previous samples of inductor current

Iborder2 = Iborder1 + m 21 td /m 2 ,

(12.2)

where I ref —the value of reference current, Δt d —the value of delay, m1 = E/L—the rising slope of inductor current; m2 = (V out − E)/L—the falling slope of inductor current; T —the period of CLOCK element (defining the switching frequency). Similar expressions for the borderlines were defined in [4]. The only problem here is to define correctly the value of m2 , as the output voltage could not be directly obtained from the discrete-time model. One of the most obvious solutions in this case is to provide a computer simulation, thus obtaining the averaged value of output voltage. However, if the reference current (defining the V out ), is chosen as the bifurcation parameter, the corresponding simulation should be provided for each value of I ref . Another solution, based on the technique provided in [11], is used by several researchers [4, 13]. This approach includes rather simple and precise formulas for calculation of V out for all three basic topologies of DC-DC converters when the I ref is known. The author of [11] introduces new variable, defining the ration of slope magnitudes: α = m 2 /m 1 .

(12.3)

Thus, the value of output voltage for the case of boost converter is equal to: Vout = E(1 + α)

(12.4)

where α itself could be found, solving the following equation from [11], modified for the inclusion of delay: (1 + α)2 E/R L + α E T /2L = Iref + td ∗ m 1 .

(12.5)

The mentioned works show that the calculated value is close enough to the values obtained from various simulation tools (PSpice and SIMULINK). In general, it is not required to calculate the V out itself, but using the definition of (12.3), the second borderline could be redefined as:

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Iborder2 = Iborder1 + m 1 td /α

(12.6)

This solution gives an opportunity not to use additional computer simulation for the definition of m2 , but to find solution of simple quadratic equation (12.5) and substituting it in (12.6). This should be done on every step, when some of the variables, included in (12.6) are changed (selected as bifurcation parameters). The overall dynamics of the system is defined in the following way: 1. if in < I border1 (see point 1 in Fig. 12.3), then system is (1): 

vn+1 = vn e−T /(R L C) i n+1 = i n + (E T )/L

(12.7)

2. if I Border1 < in < I ref (see point 2 in Fig. 12.3), then system is (2): vn+1 = e−mtoff [K 1 cos(μtoff ) + K 2 sin(μtoff )] + E  i n+1 = e

−mtoff

(K 1 cos(μtoff ) + K 2 sin(μtoff ))(1/R L − mC)



+ μC(−K 1 sin(μtoff ) + K 2 cos(μtoff ))

+ E/R L (12.8)

3. if I ref < in < I ref + m1 Δt d (see point 3 in Fig. 12.3), then there are two possible scenarios, which are dependent on some additional conditions: the development of the system in the previous cycle; as it is shown in the Fig. 12.3 the point 3 could be reached by two different ways—from point 3.1 and point 3.2, thus: 3.1. if I Border1 < in−1 < I Border2 (see point 3.1 in Fig. 12.3), then the dynamics of the system between in and in+1 is governed by (3): vn+1 = e−mT [K 3 cos(μT ) + K 4 sin(μT )] + E  i n+1 = e−mT

 (K 3 cos(μT ) + K 4 sin(μT ))(1/R L − mC) + μC(−K 3 sin(μT ) + K 4 cos(μT ))

+ E/R L (12.9)

3.2. if in−1 < I Border1 (see point 3.2 in Fig. 12.3), then the dynamics of the system between in and in+1 is governed by case (2) and defined as (12.8). The situation when I Border2 < in−1 is not observed, as in this case the value of in will be less then I ref , that is disagreement with the first defined if statement (Iref < i n < Iref + m 1 td ).

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The following designations are used in (12.7)–(12.9):   K 1 = vn e−2mton − E; K 2 = (Iref + m 1 td )/C − m(vn e−2mton + E) /μ; K 3 = vn − E; K 4 = (i n /C − m(vn + E))/μ; m 1 = E/L; ton = (Iref − i n )/m 1 + td ;  √ toff = T − ton ; m = 1/(2RC); p = 1/ LC; μ = p2 − m 2 .

In general, it should be noted that the obtained model is more complicated then the model with ideal switchings, as now for the calculation of the values in+1 it is necessary to take into account not only the position of in , but also the value of in−1 (that adds one more dimension to the model).

12.3 Results of Bifurcation Analysis On the basis of the discrete-time model defined in the previous section, the complete bifurcation analysis of the boost converter is provided. The reference current I ref is chosen as primary and the value of delay Δt d as the secondary bifurcation parameter. Parameters of the boost SPC are chosen in accordance with typical values in practical modern applications: E = 3.3 V; L = 150 µH; C = 0.5 µF; R = 40 ; Iref = 0.2 . . . 1 A; T = 10(µs); td = (0 . . . 0.2) ∗ T (s). The bifurcation diagrams are constructed by means of numerical calculation of periodic regimes, defining their stability and subsequent continuation of them, varying the primary bifurcation parameter. This approach allows the detection of all stable (dark lines in the bifurcation diagrams) and unstable (light lines in the bifurcation diagrams) periodic regimes of interest as well as investigation of possible interactions of these regimes with the borderlines, defined in (12.1) and (12.6). The brute-force approach is used in order to depict chaotic modes of operation (shaded areas in the bifurcation diagrams). First, the complete bifurcation diagram is constructed for the system without delay and then various dynamic scenarios are investigated and compared, introducing different values of Δt d . • Figure 12.4 shows the complete bifurcation diagram for Δt d = 0. This is the case of ideal switching, thus the borderlines (defined in (12.1) and (12.6)) are equal and represented in Fig. 12.4 as single dashed line. In this and the following diagrams, the inductor current sample values in are normalized with respect to I ref in order to be able to observe the slightest changes in the dynamical structure of the diagram up to scale. For small values of the reference current the SPC, as expected, operates in stable P1 regime, that loses its stability via smooth period doubling (PD) bifurcation at I ref ≈ 0.27 A. Further increase of primary bifurcation parameter leads to the chaotization of the system through classical PD cascade. Three points, defining border collisions of stable and unstable branches of the diagram are depicted as:

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Fig. 12.4. The complete bifurcation diagram for the boost converter with Δt d = 0

• BC1—non-smooth transition from stable P2 to stable P4 regime occurs; • BC2—the shape of the unstable branch is slightly changed, however the stability of the regime is not effected; • BC3—nor the shape of the branch, not the stability of the P3 regime is affected by collision with borderlines. The mentioned BC points allow ascertaining, that for the defined parameter values, in the system without delays in the control circuitry, the non-smooth effects, introduced by the presence of switching borders are negligible, not changing the structure of the diagram and having almost no effect on the stability of periodic regimes. In order to examine the effect of the control non-idealities, small value of the delay Δt d = 0.05 * T s is introduced and the complete bifurcation diagram is shown in Fig. 12.5. The constructed diagram shows that even for the small values of Δt d , the structure of dynamical transitions noticeably changes. The collision with both borderlines should be taken into account and the separation of them is defined by the value of Δt d . As in the case with ideal switching, the system still is able to operate in the stable P1 mode within small region of I ref values, and this regime loses its stability, as the characteristic multiplier crosses the unity circle through −1 (defining PD bifurcation) at I ref ≈ 0.26 A. Further, the dynamical scenario and structure of bifurcation diagram is defined by several BC points: • at BC4 P21 regime not only loses its stability, but also could not be detected after this point by means of Newton-Raphson method; the other regime P22 with the same periodicity, larger current and voltage “swing” and its own basin of attraction coexists with P21 in the vicinity of BC4, evolving through period-doubling bifurcation and disappearing at BC5;

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Fig. 12.5. The complete bifurcation diagram for the boost converter with Δt d = 0.05T

• at BC6 the I border2 cuts the stable branch of P8 regime, leading to sudden abrupt chaotization of the dynamics; thus the large scale chaotic attractor is formed by uncommon structure P8-chaos-P4-P8-chaos; • for the same I ref value as BC6, the unstable branch of P22 regime reappears; • at BC7 the unstable branch of P8 regime is cut by I border1 ; • at BC8 the stable branch of P3 regime collides with I border2 leading to sudden formation of multipeace chaotic attractor. The observations made for the system with even small values of Δt d show that the structure of bifurcation diagram changes, affecting the chaotization sequences, sudden changes in the stability and disappearance of stable and unstable periodic regimes. As it has been already mentioned, typical and practically relevant values of the delay of analog control circuitry are greater than the previously observed. So the last complete bifurcation diagram was constructed for Δt d = 0.2 * T s (see Fig. 12.6). In this case the structure of the diagram will also be described on the basis of more distinctive points. First, it should be mentioned that for the lowest observed value of the reference current the P1 solution, required for the correct operation of SPC, is not the only stable periodic regime, as it coexists with the stable P22 mode. This means that each of the regimes has its own basin of attraction and the system may intermittently change dynamics due to the presence of some unavoidable noise. It should be noted that the restriction on the minimal value of I ref = 0.2 A used during the investigation is not occasional. Simulation results show that for I ref < 0.2 A, the output voltage would be equal or even smaller than the input one, so the boost converter is no longer providing its main function—boosting the input voltage. Thus these regions should not be considered as practically relevant and are disregarded in the bifurcation diagrams.

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Fig. 12.6. The complete bifurcation diagram for the boost converter with Δt d = 0.2T

The most specific characters of the dynamics of boost converter with Δt d = 0.2 * T s are defined by following smooth and non-smooth events (see Fig. 12.6): • P1 bifurcates to stable P21 that is cut by I border2 at the point BC9; however, the chaotization of the system is governed by the development of P22 regime; the smooth PD bifurcation could be observed for I ref = 0.33 A right between two borderlines; at BC10 the collision with I border1 leads to sudden disappearance of P4 and transition to high periodic orbits and following chaos; • all periodic windows are formed by arrangement of only stable branches of periodic regimes (see for comparison Fig. 12.4, where P3 window is formed by classical Neimark-Sacker bifurcation with formation of both—stable and unstable regimes); • P5 window—first P5 regime goes through single PD bifurcation and then sudden chaotization at BC11 after collision with I border1 appears; • P8 window is involved in rather uncommon scenario: chaos—direct transition to P8 at BC12—direct transition to chaos at BC13; • periodic window starting with P3 (see BC14 in Fig. 12.6) goes through PD cascade and non-smooth transition to chaotic mode of operation (see BC15 and BC16 in Fig. 12.6); • in all cases stable and unstable regimes are cut by borderlines and could not be detected for greater values of primary bifurcation parameter: I border2 cuts the leftmost branches of periodic regimes; I border1 cuts the rightmost parts of periodic windows, preventing the development of classical PD route to chaos. The previous results show that the introduction of quite realistic delay of the control circuitry in the analysis of the dynamics of boost converter allows the detection of great variety of phenomena that are not observed with ideal current mode control. The delay affects the development of main P1 regime, causing the appearance of coexisting subharmonic modes of operation—thus making the operation of SPC unreliable

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even for small values of I ref . It could be asserted, that the delay defines bifurcation patterns of the complete bifurcation diagram influencing the development of smooth and non-smooth bifurcations and causing the appearance of different abrupt transitions from and to chaotic modes of operation within wide range of primary bifurcation parameter. In order to study the dynamical scenarios of the system for wider range of primary and secondary bifurcation parameters it is possible to construct the complete bifurcation diagrams (like those in Figs. 12.4, 12.5, and 12.6) for each value of Δt d . However, more suitable tool in this case would be the bifurcation map, depicting the regions of existence of stable periodic modes of operation, when two SPC parameters are changed. The corresponding bifurcation map for the boost converter under study is shown in Fig. 12.7. Dashed areas represent different subharmonic regimes (up to P12), and white areas represent regions of chaotic or higher periodic (>P12) modes of operation. The map clearly shows that the region of existence of stable P1 mode is diminishing for growing values of Δt d . On the other hand, the parameter space, occupied by stable P2 and P4 regimes becomes even larger for greater values of delay. As it is shown in Fig. 12.7 for relatively small values of t d < 0.1 * T the chaotic region is interrupted by relatively small amount of narrow periodic windows. However, the number and the width of mentioned windows increase with the introduction of greater delays (see e.g. the expanding P3 regime), making the chaotic region less robust.

Fig. 12.7. Bifurcation map of the boost converter under current mode control with switching delays

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12.4 Conclusions The great amount of commonly produced SPC controller integrated circuits provides the possibilities to operate under certain predefined value of reference current, predicted by engineers using of simple analytical calculations or computer simulation. However, it could be seen from the results obtained in this paper that under certain assumptions, oversimplified practical models could be irrelevant, as the delay in the control circuitry introduces some major changes in the dynamics of the system, shifting the stability border to the region of smaller values of I ref . Thus even the controllers that are inherently supposed to be stable under all possible conditions, could behave unpredictably. The complete bifurcation analysis allowed establishing the interactions of new borderlines, defined by switching non-idealities, with branches of stable and unstable periodic regimes. Constructed diagrams show that the dynamics of the SPC drastically changes at points, defined by BC with mentioned borderlines, leading to the formation of uncommon bifurcation sequences and sudden chaotization. Rather unusual non-smooth phenomenon has been observed at some BC points, when the collision with switching border leads to complete disappearance of unstable branches of bifurcation diagram and in some cases subsequent reappearance at some other BC point. This phenomenon has not been completely explored and requires detailed analysis with application of more complicated computational techniques and will be addressed in the future research. Acknowledgements Support for this work was provided by the Riga Technical University through the Scientific Research Project Competition for Young Researchers No. ZP-2014/11.

References 1. J.M. Rivas, D. Jackson, O. Leitermann, et al., Design considerations for very high frequency dc-dc converters, in Proceedings of Power Electronics Specialists Conference PESC ‘06, Jeju, South Korea (2006), pp. 1–11. https://doi.org/10.1109/pesc.2006.1712114 2. T. Nussbaumer, M. Heldwein, G. Gong, J. Kolar, Prediction techniques compensating delay times caused by digital control of a three-phase buck-type PWM rectifier system, in Proceedings of the IEEE IAS Industry Applications Conference and Fortieth IAS Annual Meeting 2 (2005), pp. 923–927. https://doi.org/10.1109/ias.2005.1518454 3. S. Choudhury, Digital Control Design and Implementation of a DSP Based High-Frequency DC-DC Switching Power Converter (Texas Instruments Inc., 2004), http://powerelectronics. com/site-files/powerelectronics.com/files/archive/powerelectronics.com/digital_power/ digital-control-design-and-implementation.pdf. Accessed 31 March 2015 4. S. Banerjee, S. Parui, A. Gupta, Dynamical effects of missed switching in current-mode controlled dc-dc converters. IEEE Trans. Circuits Syst. Part II 51(12), 649–654 (2004). https://doi. org/10.1109/TCSII.2004.838438 5. B. Kirisken, H.F. Ugurdag, Cost-benefit approach to degradation of electrolytic capacitors, in Proceedings of Annual Reliability and Maintainability Symposium (RAMS), Colorado Springs, USA (2014), pp. 1–6. https://doi.org/10.1109/rams.2014.6798436

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6. H. Asahara, T. Kousaka, S. Banerjee, Stability analysis in the current-mode controlled DC/DC buck converters with switching delay, in Proceedings of the Nonlinear Dynamics of Electronic Systems Conference NDES 2012, Wolfebuttel, Germany (2012), pp. 1–4. ISBN: 978-3-80073444-3 7. H. Asahara, I. Yutaka, T. Yusuke et al., Effect of time lag in response to switching signal in interrupted electric circuit. Circuits Syst. Signal Process. 33(9), 2695–2707 (2014). https://doi. org/10.1007/s00034-014-9780-y 8. F. Vasca, L. Iannelli (eds.), Dynamics and Control of Switching Electronic Systems (Springer, London, UK, 2012). ISBN 978-1-4471-2884-7, https://doi.org/10.1007/978-1-4471-2885-4 9. M.M. Al-Hindawi, A. Abusorrah, Y. Al-Turki et al., Nonlinear dynamics and bifurcation analysis of a boost converter for battery charging in photovoltaic applications. Int. J. Bifurc. Chaos 24(11), 1450142 (2014). https://doi.org/10.1142/S0218127414501429 10. C. Basso, Switch-Mode Power Supplies Spice Simulations and Practical Designs (McGraw-Hill Inc., New York, USA, 2008). ISBN: 978-0071508582 11. S. Banerjee, G.C. Verghese (eds.), Nonlinear Phenomena in Power Electronics. Attractors, Bifurcations, Chaos, and Nonlinear Control (Wiley-IEEE Press, 2001). ISBN: 978-0-78035383-1 12. C.K. Tse, Complex Behavior of Switching Power Converters (CRC Press, Boca Raton, US, 2003). ISBN: 978-0-8493-1862-7 13. R. Pollanen, A. Tariainen, O. Pyrhonnen, Simulation of a current-mode DC-DC boost converter in chaotic regime evaluating different simulation methods. Electr. Eng. 88(1), 35–44 (2005). https://doi.org/10.1007/s00202-004-0259-x

Part IV

Econophysics

Chapter 13

Complex Network Time Series Analysis of a Macroeconomic Model I. P. Antoniades, S. G. Stavrinides, M. P. Hanias and L. Magafas

Abstract Time series produced by an idealized macro-economic model are analyzed by means of conversion to complex networks by three different methods: the recursive graph method, the natural visibility graph method and the ordinal partition graph method. For several values of one of the model’s control parameters yielding both fully chaotic time-series and intermittent chaos very close to periodicity the corresponding complex networks are obtained for each conversion algorithm, and several network metrics are evaluated: average degree, number of nodes, connectivity ratio, characteristic path length, clustering coefficient, assortativity and local dimensionality. Moreover, multifractal analysis of the chaotic time series is performed which reveals the multifractal structure of the orbits in phase space for the chaotic time series. This multifractal structure is also depicted in the scale-free nature of the corresponding complex networks which agrees with literature results. Most metrics clearly distinguish between fully chaotic and intermittent time series. Some metrics have significantly different values for chaotic series obtained at different values of the control parameter revealing subtle differences in phase space structure and system dynamics. The average degree of the ordinal partition networks together with assortativity showed that complex network time series analysis captures the repetition of particular patterns in the original time series and particular temporal correlations that would not be easy or even possible to capture with traditional nonlinear analysis methods. Finally, the Lyapunov exponent of the time series is shown to be linearly correlated with the average local dimensionality metric of the complex network obtained by the visibility graph method, a result that is reported for the first time.

I. P. Antoniades Division of Science & Technology, American College of Thessaloniki, Thessaloniki, Greece S. G. Stavrinides (B) School of Science and Technology, International Hellenic University, Thessaloniki, Greece e-mail: [email protected] M. P. Hanias · L. Magafas Physics Department, International Hellenic University, Kavala, Greece © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_13

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13.1 Introduction Physical modeling of macro- and micro-economic systems are part of the relatively new field of Econophysics that has attracted significant research activity in the past decade. Modeling a time-evolving economic system, which in most cases is particularly complex, in order to understand the underlying dynamics, classify similar systems and predict their future evolution based on knowledge of the past, is one of the most interesting research domains, within this area.

13.1.1 The Vosvrda Idealized Macro-economic Model One of the methods is describing the dynamics of an economic system by sets of differential equations [1, 2]. Such a case is the Vosvrda Idealized Macroeconomic Model with foreign capital investment [2, 3], which is described by a set of three first order nonlinear differential equations involving the following dynamic variables: S(t) that represents the savings of households, Y (t) that represents the Gross Domestic Product (GDP) and F(t) that represents the foreign capital inflow, all at year t. dS = aY + pS(k − Y 2 ) dt

(13.1a)

dY = u(S + F) dt

(13.1b)

dF = mS − rY dt

(13.1c)

a is the variation of the marginal propensity to savings, p is the ratio of the capitalized profit, u is the output/capital ratio, k is the potential GDP (it can be set to 1, as a unit of GDP—Y, S, F then represent the percentage part of the potential GDP), m is the capital inflow/savings ratio and r is the debt refund/output ratio. For certain values of these parameters the solution reaches a stable steady state, for others it shows periodic behavior and for others it is chaotic [2].

13.1.2 Complex Network Time Series Analysis On the other hand, a very interesting method to analyze time series of any complex system that has been proposed a while ago [4], involves converting a time series into a complex network, i.e. a graph whose nodes represent particular points or patterns of the time series, while connections represent particular relationships between any pair of nodes, according to a predetermined rule (algorithm). Once the series is

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converted into a network, then well-known network metrics originating from graph theory can be calculated, providing further insight into the underlying dynamics of the complex system and revealing subtle details of the phase-space structure. In this sense, converting a time series into a network and studying it by utilizing graphtheory metrics, introduces an additional tool for characterizing and understanding complex systems, providing with a fingerprint of the dynamical behavior, additional to traditional nonlinear analysis methods. Several algorithms have been proposed in the literature for converting a time series into a network [4–8]. In the present work, we use three particular algorithms: (1) the Recurrence Graph algorithm (RG) [4], (2) the Natural Visibility-Graph algorithm (NVG) [7] and (3) the Ordinal Partition Graph algorithm (OPG) [8]. In the original RG algorithm, a discrete scalar time series xi ∈ R is first processed and  obtained in the proper a re-constructed vector time-series xi = x1i , x2i , . . . , xdi embedding dimension d of the dynamical system by a method such as Principal Component Analysis (PCA) [9]. In the reconstructed d-dimensional phase-space each vector xi corresponds to a node in the network. Any two nodes i and j are connected, if the corresponding phase-space points xi and xj are separated by a Euclidean distance s < ε, where ε is some arbitrary threshold value. In a modified version of this algorithm, each point in the time series is examined and a fixed number k of its nearest neighbors in phase space (again determined by Euclidean distance) corresponds to the nodes to which it will be connected. This is the version used in particular previous studies (for instance [10]), its advantage being that the total number of edges and the average number of connections per node (average degree of network) is fixed and no threshold value for distance s is needed. This is the version we use in the present work. In the NVG method, no phase-space embedding is needed. Each point in the original time series corresponds to a network node. Any two nodes i and j are connected if the corresponding series values x i and x j are “visible” from each other, i.e. the slope of the straight line connecting x i and x j is smaller than the slope of the straight lines connecting x i to every point k lying between i and j. In Fig. 13.1, we show an example of ‘visibility’ connections between pairs of points in a particular time series. Black arrows depict points that are visible from each other and are thus connected and red arrows show pairs of points that are not visible from each other and are not connected. Finally, the OPG algorithm captures systems dynamics rather series geometry, following a different approach. First, the time series {x i } is divided into partitions consisting of w consecutive series points. In this way a series of length L, is divided into L − w + 1 partitions. Each partition {x i , x i+1 , x i+2 , …, x i+w } is then ordered in terms of the values of x. The ordered set of the indices {1, 2, 3, … w} of each partition (ordinal) corresponds to a unique network node. Finally, two nodes i and j are connected by an edge, if the partition corresponding to ordinal j occurs just after the partition corresponding to ordinal i in the time series. An example, of how an OPG is constructed is shown in Fig. 13.2. Several other time series to network conversion algorithms as well as variations of the ones mentioned above have appeared in literature in recent years and have

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Fig. 13.1 Example of how an NVG algorithm works. Black arrows show pairs of series points that would be connected by an edge and red arrows show points that would not

Fig. 13.2 Example of how an OPG is constructed. Here the length of each partition is 4. Figure is taken with permission from [8]

been applied in both model and real time series from various disciplines such as economics (stock market), biology (human EEG), physics (turbulent flows), etc. A concise review of known algorithms and real applications can be found in [11] and references therein. Once a time series is converted to a network, well-known network metrics, such as, average degree, connectivity, characteristic path length, assortativity, modularity, local and global dimensionality can be calculated. Some metrics are global, i.e. is have one value for the whole graph, and others are defined locally, i.e.. For local measures, one usually looks at a histogram of values for the entire network or the average value and other statistical quantities. The values of such metrics characterize various aspects of the topology of the network, e.g. whether it is normal or scale-free, the degree of clustering etc. The structure of the network itself is connected to the topological and dynamical properties of the time series depending on the algorithm used. For example, it has been shown that fractal time series yield

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scale-free networks for both the RG and NVG algorithms. The values of other metrics can be used to distinguish between chaotic, semi-periodic or periodic behaviors of the time series but also to distinguish among chaotic time series catching subtle differences in the underlying dynamics. A thorough study on how this is achieved can be found in [12]. In the present study, the following network metrics were used: (1) degree ki , which is simply the number of connections initiating from a particular node i. Both average degree and degree distributions are reported. (2) Local clustering coefficient C i [13], which is defined as the proportion of the connections among all nearest neighbors k ij of a node i, that are also connected to each other, over the total number of possible connections among the nearest neighbors. A zero value of C i denotes that no connections exist among adjacent nodes (nearest neighbors) of node i, and a value equal to 1 (maximum) denotes that all possible connections among the nearest neighbors of i exist. (3) Assortativity coefficient r, which is the Pearson correlation coefficient of degree between pairs of connected nodes [14]. Positive values of r indicate a correlation between nodes of similar degree, while negative values indicate a correlation between nodes of different degree. In general, r lies between −1 and 1. For example, if nodes of high degree tend to be adjacent to nodes of also high degree, assortativity coefficient is positive, and if nodes of high degree tend to be adjacent mostly to nodes of low degree, then assortativity coefficient is negative. (4) Local dimensionality Di [15], which is defined as follows: Taking the number of nodes within a neighborhood of node i, as a function of distance from node i (distance being defined as the least number of edges that need to be transversed to go from node i to a node j), and assuming this function is well described by a power law with respect to distance, then the power law exponent is Di . (5) Characteristic (or average) path length L, which is defined as the distance between any nodes i and j averaged over all pairs i, j. (6) Number of nodes N, (7) Connectivity ratio C r , which is defined as the ratio of total number of edges in the graph to the total number of possible edges among all nodes in the graph. The last two metrics are not meaningful for the RG algorithm, as the total number of nodes and connections of the obtained networks are fixed. The total number of nodes is not meaningful for the NVG algorithm networks either, as it is also fixed and equal to the number of series points. In this study, we report the average values and standard deviations of the metrics that are locally defined. We also report the degree distributions, which suffice to define the type of network structure, for example if it is a random, or scalefree, or small-world network. Interesting conclusions about structural characteristics of networks can in principle be drawn by examining the distributions of the other local metrics, but these are not reported here due to lack of space.

13.2 Computations and Results The Vosvrda model system was numerically integrated using a Runge-Kutta method with MATLAB™ ode45 solver.

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Fig. 13.3 Bifurcation diagram for S (savings) as a function of control parameter a. The values of the other control parameters are: p = 0.01, k = 1, u = 0.031847, m = 0.2, r = 0.25

A full-scale scan of the system behavior in terms of the control parameter a was performed and is illustrated in the bifurcation diagram of variable S (savings) as shown in Fig. 13.3. In all calculations, all other model control parameters are held fixed at the following values: p = 0.01, k = 1, u = 0.031847, m = 0.2 and r = 0.25 as in [16]. Both period-doubling and intermittency routes to chaos are clearly present. Next, for several values of a, time series for S, F and Y were obtained and converted to complex networks with each of the three algorithms presented in the previous section. For the RG algorithm, the reconstructed phase-space series of S were first obtained. A delay window of 10 series points was used for the Takens [17] embedding, to ensure the numerical integration time step does not affect the system dynamics. The embedding dimension is equal to 3, as expected. For each node a fixed number of 4 nearest neighbors were connected. For the NVG and OPG algorithms, the discrete time series of all local extrema (maxima and minima) of the original time series were used. For each conversion algorithm the same length series was used for every value of the control parameter. Figure 13.4 shows the resulting complex networks obtained from an intermittent chaotic time series very close to period-8 region for a = 0.113 (a, b, e, f) and a fully chaotic time series for a = 0.07 (c, d, g, h) for both the RG algorithm [(a)–(d)] and the NVG algorithm [(e)–(h)]. The figures on the right column are blowups of the boxed regions in each figure on the left. The fully chaotic time series clearly yield more complex and self-similar looking network structures. Comparing the fully chaotic and intermittent chaos-period 8 time series RG’s (Figs. 13.4a, b and 13.5c, d), it can be seen that, although the fully chaotic time series yield a more complex network, complexity is also present in the intermittent time series network. It is reminded that, as also mentioned in Sect. 13.1.2, the RG algorithm used in this study yields networks with the same number of connections irrespective of the time series. The complexity of the network, therefore, depends only on higher order metrics such as degree distributions, clustering coefficient, assortativity etc. The RG graph is more sensitive to presence of even small degree of chaos. This is expected because even the slightest disturbance of the distance between points in phase space will alter the nearest neighbor profile upon which the construction of this type of network depends. On the other hand, the NVG method distinguishes more clearly between fully chaotic and close-to-periodic orbits, as can be seen by comparing the graphs in Fig. 13.4e–h. The periodicity present in the first series (Fig. 13.4e, f)

NVG

RG

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 13.4 Complex networks obtained from a chaotic time series for a = 0.07 (c, d, g, h) and intermittent chaos-period 8 time series for a = 0.113 (a, b, e, f). a–d correspond to RG algorithm and e–h to NVG algorithm. b, d, f and h are blow-ups of the boxed regions in a, c, e and g in respect

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Fig. 13.5 Complex networks obtained for a chaotic time series for a a = 0.113 (intermittent chaos-period 8 time series) and b for a = 0.07 with the OPG algorithm

4 3

Fig. 13.6 Average local dimensionality of NVG networks versus the respective time series Lyapunov exponent λ

2 1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

λ is clearly producing a repetitive structure in the NVG graph, especially shown in Fig. 13.4f, whereas the fully chaotic time series produces a self-similar network. The respective graphs for the OPG algorithm, one capturing the dynamics rather than the geometry of the time series, are shown in Fig. 13.5. For the construction of the network, the series was partitioned using a window of length w = 8, in order to match the period for the intermittent chaos time series close to period-8. For this time series, the network is trivial (Fig. 13.6a): it contains only 8 nodes (equal to the number of distinct ordinals), and no edges, only self-connections, i.e. each node is only connected to itself (self-connections are not shown in the figure). The weak presence of chaos is not affecting the order of the values of the series within each window. Thus are only 8 distinct ordinals, each one following itself in time. The fully chaotic time series, however (Fig. 13.5b), also constructed by a w = 8 length window, shows a much more complex structure. In Table 13.1 the values of several network metrics applied to six RG networks are shown. The values of the control parameter a is shown in column 1. In Tables 13.2 and 13.3 network metrics for the NVG and OPG algorithm networks are shown in respect. We notice that almost all metrics clearly distinguish the intermittent chaos-period 8 time series from the chaotic ones. Several metrics have clearly different values when comparing different between chaotic time-series, depending on conversion algorithm too. The Lyapunov exponent of the respective time series is also shown in Table 13.1. For some metrics of networks obtained from the fully chaotic time series, there is no apparent correlation with the respective value of the Lyapunov exponent, but for other metrics the contrary occurs.

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Table 13.1 Results of metrics for the RG algorithm networks a

Type of time series

λ

Ci 

l

r

0.100

Chaos

0.4372

0.3854

3.8371

0.2137

0.070

Chaos

0.4783

0.3905

4.0764

0.2031

0.113

Intermittent chaos (period 8)

0.1416

0.6581

2.189

0.3978

0.130

Chaos

0.3794

0.4373

3.2528

0.1924

0.045

Chaos

0.4264

0.3981

3.4982

0.2703

0.120

Chaos

0.4023

0.4398

3.4557

0.2199

Table 13.2 Results of metrics for the NVG algorithm networks a

Type of time series

C r (×10−3 )

Di 

ki 

Ci 

0.100

Chaos

1.38

3.28

4.75

0.724

0.070

Chaos

1.72

3.60

4.72

0.113

Intermittent chaos (period 8)

1.09

1.68

4.12

0.130

Chaos

1.13

3.00

0.045

Chaos

1.54

3.19

0.120

Chaos

1.13

3.07

l

r 6.58

0.199

0.722

6.93

0.185

0.705

43.29

4.74

0.722

7.28

0.167

4.87

0.734

6.54

0.157

4.62

0.724

7.56

0.124

0

Table 13.3 Results of metrics for the OPG algorithm networks for w = 8 a

Type of time series

N

Cr

ki 

Ci 

l

r

0.100

Chaos

405

0.058

3.24

0.3297

0.6709

−0.341

0.070

Chaos

705

0.042

7.1574

0.2184

0.7103

−0.132

0.113

Intermittent chaos (period 8)

8

0

0

0

0

0

0.130

Chaos

218

0.061

1.4389

0.2181

0.8148

−0.197

0.045

Chaos

176

0.075

1.5107

0.2964

0.7221

−0.315

0.120

Chaos

304

0.062

2.1038

0.3365

0.6985

−0.282

For instance, the average local dimensionality of the NVGs (Table 13.2), which was computed by averaging the local dimensionality values only of those nodes for which the Pearson’s R value of the least-χ 2 power law fit is greater than 0.95, appears to be linearly correlated to the value of the Lyapunov exponent λ, as shown in Fig. 13.6. This correlation means that, as the Lyapunov exponent is a measure of the rate of divergence of nearby orbits, the higher this rate, the faster the rate of growth of the number neighbors of a node as a function of distance from the node. This result can be justified by the fact that a large Lyapunov exponent in general results in a more complex structure of the time series, therefore, all the more series points j are “visible” from a particular point i and successively, more nodes are visible from

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each series point j, a.s.o. Therefore, the number of neighbors grows faster at each successive neighborhood of node i in networks created by the NVG algorithm. It is interesting to notice that the clustering coefficient of the NVG networks does not vary significantly with control parameter a, with values slightly above 0.70, even for the (almost) periodic series. Average degree is also almost independent of a. The relatively high clustering coefficient suggests that time series points (comprising of the original time series local extrema) that are ‘seen’ from a series point, most likely also ‘see’ each other. It is also interesting that the average local dimensionality  D i  is close to three, approximately equal to the embedding dimension of the series with the exception of the network corresponding to the intermittent series close to period 8 (a = 0.113) whose average local dimensionality is much lower. For this series, the network is much more ‘1-dimensional’, it resembles a line, and thus the characteristic path length is naturally much higher than in the chaotic time series. Also, assortativity is exactly zero, which means that nodes of high degree are equally likely to be adjacent to nodes of high degree or low degree. The values of assortativity of the other series are slightly positive. The OPG networks, as said, are different from the other two types because connections between nodes depend on the dynamics of the time series rather than the topology of phase space. It is interesting to note that, the number of nodes N in the OPG network represent the number of distinct length 8 ordinals present in the time series. The maximum number of allowed permutations of an ordinal of length 8 is 8! = 40,320. However, the number of nodes in the OPG in Fig. 13.5b is only 705 for the network corresponding to a = 0.07 and it is much less for the other chaotic time series. This implies that there are only a few ordinal patterns present in the chaotic time series which represents a clear separation from randomness. Connectivity ratio is also quite low. This implies that the limited number of ordinal patterns present in the time series are, in addition, causally connected to a very small set of other ordinal patterns. This is also evident by the relatively small average degree of the networks. It is particularly interesting to notice that the average degree of chaotic series for a = 0.13 and a = 0.045 is surprisingly low ( 2 2i , with the initial conditions ai (n) =  n2  for 1 ≤ n ≤ 2 2i . ai (n + 1) − ai (n) ∈ {0, 1} for all n, i ≥ 1 and ai (n) hits every positive integer. Proof Firstly, let us see ai (2 2i  + 1) by definition, that is, ai (2 2i  + 1) = 2 2i  + 1 − ai (ai (2 2i  + 1 − i)) − ai (2 2i  + 1 − ai (2 2i  + 1 − i)). In here if i is even, ai (i + 1) = i + 1 − ai (ai (1)) − ai (i + 1 − ai (1)) = i + 1 − 1 − ai (i) = 2i = ai (i). If i is odd, ai (i + 2) = i + 2 − ai (ai (2)) − ai (i + 2 − ai (2)) = i+1 = ai (i + 1). So ai (2 2i  + 1) = = i + 2 − 1 − ai (i + 1) = i + 1 − i+1 2 2 i ai (2 2 ) for all i ≥ 1. Since we know that ai (t) − ai (t − 1) ∈ {0, 1} for all 2 ≤ t ≤ 2 2i  + 1, we can proceed by induction as below. We must show that ai (n + 1) − ai (n) ∈ {0, 1} for all n ≥ 2 2i  + 1. By definition equations are below.

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ai (n + 1) = n + 1 − ai (ai (n − i + 1)) − ai (n + 1 − ai (n − i + 1))

(14.1)

ai (n)

(14.2)

= n − ai (ai (n − i)) − ai (n − ai (n − i))

From (14.1) and (14.2), ai (n + 1) − ai (n) = n + 1 − ai (ai (n − i + 1)) − ai (n + 1 − ai (n − i + 1)) − (n − ai (ai (n − i)) − ai (n − ai (n − i))) = 1 − (ai (ai (n − i + 1)) − ai (ai (n − i))) − (ai (n + 1 − ai (n − i + 1)) − ai (n − ai (n − i))).

Case 1. ai (n − i + 1) = ai (n − i) + 1. At this case, ai (n + 1) − ai (n) = 1 − (ai (ai (n − i + 1)) − ai (ai (n − i))) − (ai (n + 1 − (ai (n − i) + 1)) − ai (n − ai (n − i))). = 1 − (ai (ai (n − i) + 1) − ai (ai (n − i))) ∈ {0, 1}.

Case 2. ai (n − i + 1) = ai (n − i). At this case, ai (n + 1) − ai (n) = 1 − (ai (ai (n − i)) − ai (ai (n − i))) − (ai (n + 1 − ai (n − i)) − ai (n − ai (n − i))). = 1 − (ai (n + 1 − ai (n − i)) − ai (n − ai (n − i))) ∈ {0, 1}. This completes the induction about slowness of ai (n) for all i ≥ 1. For second part of our proof let us assume that ai (n) = K i is the maximum value of sequence and Ni is the first occurence K i . So, ai (Ni + ti ) = K i for all ti ≥ 0. At this case ai (Ni + ti ) = Ni + ti − ai (ai (Ni + ti − i)) − ai (Ni + ti − ai (Ni + ti − i)) = Ni + ti − ai (K i ) − ai (Ni + ti − K i ) for all ti ≥ i. If we choose ti ≥ K i then 2K i = Ni + ti − ai (K i ) and this is contradiction. Since ai (n) is slow, ai (n) must hit every positive integer. See Fig. 14.2 for illustration of first five members of this family. Although we completely prove the existence of infinitely many different solutions, we lose symmetry for i > 1 in their generational structures. See Fig. 14.3 to a nice connection between a1 (n) and c(n) which is HofstadterConway $10000 sequence. a1 (n) and c(n) behave in a complete relationship. A new auxiliary method shows the connection in their generational structures from another perspective. To this aim, Let b(n) = n − b(c(n)) − b(n − c(n)) and t (n) = n − t (a1 (n)) − t (n − a1 (n)) with b(1) = b(2) = t (1) = t (2) = 1. Both sequences have same generational boundaries. See Fig. 14.4 for main idea of corresponding method and intersecting behaviors. Additionally, there are still possibilities for different symmetric solutions especially for case i = 1. To this aim, we propose another sequence family as follows.

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Fig. 14.2 s1 (n) : black, s2 (n) : red, s3 (n) : yellow, s4 (n) : green, s5 (n) : blue where si (n) = ai (n) − n2 for n ≤ 210

Fig. 14.3 Scatterplots of c(n) −

n 2

(red) and a1 (n) −

n 2

(black) for n ≤ 210

14 On Families of Solutions for Meta-Fibonacci Recursions …

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Fig. 14.4 Plots of 2b(n) − n which is A317754 in OEIS (red) and 2t (n) − n which is A317854 in OEIS (black) for n ≤ 215

Proposition 14.2 Let ai∗ (n) = n − ai∗ (ai∗ (n − 1)) − ai∗ (n − ai∗ (n − 1)) for n > 2i , with the initial conditions ai∗ (n) =  n2  for 1 ≤ n ≤ 2i . ai∗ (n + 1) − ai∗ (n) ∈ {0, 1} for all n, i ≥ 1 and ai∗ (n) hits every positive integer. Proof Similar proof can be constructed with previous proposition since a1∗ (n) = a1 (n). See Fig. 14.5 for illustration of curious generational structures which powers of 2 are determinative for main boundaries. For Newman generalization on Hofstadter-Conway $10000 sequence, Kubo and Vakil proved the asymptotic properties based on largest root of corresponding characteristic polynomial [9]. Based on this fact, we propose a new generalization as follows. Theorem 14.1 Let ai, j (n) = ai, j (ai, j (n − j)) + ai,j (n −ai, j (n − 1)) for n > 2(i − 1) + 3φi , with the initial conditions ai, j (n) = √ 1+ 5 2

n+2 1+φ

for 1 ≤ n ≤ 2(i − 1) +

3φi where φ = and j ∈ {1, 2}. ai, j (n + 1) − ai, j (n) ∈ {0, 1} for all n, i ≥ 1 and ai, j (n) hits every positive integer. Proof In here if j = 1, proof of slowness is easy by induction since our initial condition function provides basis to this attempt. More precisely, ai,1 (t + 1) = ai,1 (ai,1 (t)) + ai,1 (t + 1 − ai,1 (t)) means that ai,1 (t + 1) − ai,1 (t) = 1 and ai,1 (t) − ai,1 (t − 1) = 0 where t = 2(i − 1) + 3φi , with the initial conditions ai,1 (n) =

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Fig. 14.5 s1∗ (n) : red, s2∗ (n) : yellow, s3∗ (n) : green, s4∗ (n) : blue, s5∗ (n) : purple where si∗ (n) = ai∗ (n) − n2 for n ≤ 210





√

for 1 ≤ n ≤ 2(i − 1) + 3φi and φ = 1+2 5 . However, similar proof necessitates additional work if j = 2. In order to prevent notational complexity, we can use a1,2 (n) = a(n). Let us assume that a(k) − a(k − 1) ∈ {0, 1} for all 2 ≤ k ≤ n. Additionally, let us assume that there is no two consecutive 1’s in sequence of a(k) − a(k − 1) for all 2 ≤ k ≤ n. We know that these are correct for small n and we proceed by induction. We must show that a(n + 1) − a(n) ∈ {0, 1} for all possible cases with conservation of our second assumption. By definition equations are below. n+2 1+φ

a(n + 1) = a(a(n − 1)) + a(n + 1 − a(n))

(14.3)

= a(a(n − 2)) + a(n − a(n − 1)) a(n) a(n − 1) = a(a(n − 3)) + a(n − 1 − a(n − 2)) a(n − 2) = a(a(n − 4)) + a(n − 2 − a(n − 3))

(14.4) (14.5) (14.6)

From (14.3) and (14.4), a(n + 1) − a(n) = a(a(n − 1)) − a(a(n − 2)) + a(n + 1 − a(n)) − a(n − a(n − 1)).

Case 1. a(n) = a(n − 1) + 1 and a(n − 1) = a(n − 2). At this case, a(n + 1) − a(n) = a(a(n − 1)) − a(a(n − 2)) + a(n + 1 − a(n)) − a(n − a(n − 1)) = a(n + 1 − (a(n − 1) + 1) − a(n − a(n − 1)) = 0.

14 On Families of Solutions for Meta-Fibonacci Recursions …

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Case 2. a(n) = a(n − 1) and a(n − 1) = a(n − 2). At this case, a(n + 1) − a(n) = a(a(n − 1)) − a(a(n − 2)) + a(n + 1 − a(n)) − a(n − a(n − 1)) = a(n + 1 − a(n)) − a(n − a(n)) ∈ {0, 1}.

Case 3. a(n) = a(n − 1) and a(n − 1) = a(n − 2) + 1. At this case, a(n + 1) − a(n) = a(a(n − 1)) − a(a(n − 2)) + a(n + 1 − a(n)) − a(n − a(n − 1)) = a(a(n − 1)) − a(a(n − 1) − 1) + a(n + 1 − a(n)) − a(n − a(n)).

We have 2 different possibilities by our initial assumptions. From (14.5) and (14.6), Case 3.1. a(n − 2) = a(n − 3) = a(n − 4). a(n − 1) − a(n − 2) = a(a(n − 3)) − a(a(n − 4)) + a(n − 1 − a(n − 2)) − a(n − 2 − a(n − 3)) = a(n − 1 − a(n − 2)) − a(n − 2 − a(n − 3)) = a(n − 1 − (a(n) − 1)) − a(n − 2 − (a(n) − 1)) = a(n − a(n)) − a(n − 1 − a(n)) = 1.

According to our second assumption, a(n + 1 − a(n)) − a(n − a(n)) = 0 because of a(n − a(n)) − a(n − 1 − a(n)) = 1, and this means that a(n + 1) − a(n) = a(a(n − 1)) − a(a(n − 1) − 1) ∈ {0, 1}. Case 3.2. a(n − 2) = a(n − 3) and a(n − 3) = a(n − 4) + 1. a(n − 1) − a(n − 2) = a(a(n − 3)) − a(a(n − 4)) + a(n − 1 − a(n − 2)) − a(n − 2 − a(n − 3)) = a(a(n − 3)) − a(a(n − 4)) + a(n − 1 − (a(n) − 1)) − a(n − 2 − (a(n) − 1)) = a(a(n − 3)) − a(a(n − 4)) + a(n − a(n)) − a(n − 1 − a(n)) = 1.

In here, a(a(n − 3)) − a(a(n − 4)) = 1 or a(n − a(n)) − a(n − 1 − a(n)) = 1. From Case 3.1., we know the result of a(n − a(n)) − a(n − 1 − a(n)) = 1. So let us consider a(a(n − 3)) − a(a(n − 4)) = 1, that is a(a(n − 2)) − a(a(n − 2) − 1) = 1 and this means that a(a(n − 2) + 1) − a(a(n − 2)) = a(a(n − 1)) − a(a(n − 2)) = 0. So again a(n + 1) − a(n) = a(n + 1 − a(n)) − a(n − a(n − 1)) ∈ {0, 1}. Now we must be sure the continuity of situation that our second assumption says there is no two consecutive 1’s in sequence of a(k) − a(k − 1) for all 2 ≤ k ≤ n. For all possible cases that we work above guarantees that if we have 1 in sequence of a(k) − a(k − 1), following first difference must be 0. That means that we will never have two consecutive 1’s in sequence of a(k) − a(k − 1) according to our

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    n+2 n+2 (black) and ai,2 (n) − 1+φ (red) for F(19) ≤ n ≤ F(20) Fig. 14.6 Plots of ai,1 (n) − 1+φ and 1 ≤ i ≤ 8 where F(n) is nth Fibonacci number

second assumption which will be preserved for all cases. This definitely completes the induction for a(n) which is a1,2 (n). For othermembers of family, it is easy to n+2 satisfies our second assumption see that initial conditions function ai,2 (n) = 1+φ for all n. In other  words, there is no consecutive 1’s in sequence of first differn+2 . Additionally, ai,2 (t + 1) = ai,2 (ai,2 (t − 1)) + ai,2 (t + 1 − ai,2 (t)) ences for 1+φ means that ai,2 (t + 1) − ai,2 (t) = 1 and ai,2 (t) − ai,2 (t −  1) = 0 where t = 2(i − n+2 1) + 3φi , with the initial conditions ai,2 (n) = 1+φ for 1 ≤ n ≤ 2(i − 1) + √

3φi and φ = 1+2 5 . With similar applying of recurrence, ai,2 (t + 2) − ai,2 (t + 1) = 0, ai,2 (t + 3) − ai,2 (t + 2) = 1, ai,2 (t + 4) − ai,2 (t + 3) = 0. So we can apply same proof of slowness with this initial condition formulation for all members of ai,2 (n) family. For second part of proof, same approach with proof of first proposition can be easily adopted and this simply shows that ai, j (n) is unbounded and it hits every positive integer due to slowness. See Fig. 14.6 for an illustration of this generalization to initial members. Since Fibonacci numbers determine the generational boundaries of corresponding humps, illustration is particularly interesting with the effect of golden ratio.

14 On Families of Solutions for Meta-Fibonacci Recursions …

Fig. 14.7 Plots of Fibonacci number

a2,1 (n) a1,1 (n) n , n

and

a2,2 (n) a1,2 (n) n , n

157

for F(8) ≤ n ≤ F(27) where F(n) is nth

Since we analyse our recurrences based on generalization which use asymptotic properties of recurrences and selected solutions, now we can investigate sensitivity of initial conditions selections. See Fig. 14.7 in order to observe a fascinating behaviour for first two members of our generalization for both alternatives. During their generational  evolution, common asymptotic properties attract the scatterings to  1 1 . , 1+φ point 1+φ

14.3 Conclusion We apply a reasonable approach for two strange meta-Fibonacci recursions related to Hofstadter-Conway $10000 sequence. We construct infinitely many theoretical solutions with curious generational structures and we also observe interesting results in terms of their sensitive dependence on initial conditions. While this work provides meaningful results in order to overcome some difficulties of analysis of metaFibonacci recursions which is intriguing subfamily of nonlinear recurrences, we hope that these attempts will be meaningful for future communications in this research field. Acknowledgements We would like to thank Prof. Steve Tanny for his helpful communication. We would also like to thank Prof. Robert Israel regarding his valuable help for Maple related requirements of this study. Finally, we would like to thank Prof. N.J.A. Sloane in particular since this work would not be possible without OEIS.

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References 1. N. Fox, An Exploration of Nested Recurrences Using Experimental Mathematics. Ph.D. Thesis, Department of Mathematics, Rutgers University (2017) 2. A.S. Fraenkel, Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups. Trans. Am. Math. Soc. 341, 639–664 (1994) 3. S.M. Tanny, A well-behaved cousin of the Hofstadter sequence. Discret. Math. 105, 227–239 (1992) 4. A. Alkan, N. Fox, O.O. Aybar, On hofstadter heart sequences. Complexity 1–8, (2017) 5. A. Alkan, On a generalization of hofstadters q-sequence: a family of chaotic generational structures. Complexity 1–8, (2018) 6. K. Pinn, A chaotic cousin of conways recursive sequence. Exp. Math. 9, 55–66 (2000) 7. K. Pinn, Order and chaos in Hofstadters Q(n) sequence. Complexity 4, 41–46 (1999) 8. N.J.A. Sloane, OEIS Foundation Inc. The On-Line Encyclopedia of Integer Sequences (2019) 9. T. Kubo, R. Vakil, On conways recursive sequence. Discret. Math. 152, 225–252 (1996) 10. C.L. Mallows, Conways challenge sequence. Am. Math. Mon. 98, 5–20 (1991) 11. C.A. Pickover, The crying of fractal batrachion 1489. Comput. Graph. 19, 611–615 (1995) 12. A. Alkan, On a conjecture about generalized Q-recurrence. Open Math. 16, 1490–1500 (2018)

Chapter 15

Chaotic Solutions for Asset Management Complexity G. Cziráki

Abstract The research introduces an extraordinary scientific discovery that is a combination of chaos theory, system theory, and finance. According to the perspective viewpoint of Bertalanffy’s General System Theory, this interdisciplinary research introduces the nonlinear model concept from other disciplines based on the isomorphism of scientific laws in the financial field. The study model empirically demonstrates isomorphism establishment and, according to Ashby’s simplification theory, possesses a key heuristic function (Ashby in Views on General Systems Theory. Wiley, New York, pp. 165–169, 1964 [1]).

15.1 Introduction In a pioneering manner, I have succeeded in making the complexity of the Mandelbrot set measurable, and I illustrate this in a simplistic and perspectivist way. The discovery involves combining Mintzberg’s [2] organizational structure configuration by correcting and refining this organizational typology model and accurately depicting the distortion, the dominance, and the determinative living space of the organization. The purpose of the system theory of economic activities is to illuminate and understand the environmental conditions for making a decision. The discovery in this study makes tangible the bounded range (frame), within which a complex system makes its decisions. This innovation aims to create an easier decision-making process by presenting a model and asset management process that simultaneously incorporates Swensen’s [3] diversification and Warren Buffet’s focusing strategy [4].

G. Cziráki (B) Alexandre Lamfalussy Faculty of Economics, University of Sopron, Sopron, Hungary e-mail: [email protected] © Springer Nature Switzerland AG 2020 S. G. Stavrinides and M. Ozer (eds.), Chaos and Complex Systems, Springer Proceedings in Complexity, https://doi.org/10.1007/978-3-030-35441-1_15

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15.2 Research Method As an example, the present study uses a model-based complex system, in this case an investment portfolio, which I have synthesized from the achievements of Mandelbrot [5] and Mintzberg—the M&M Model (Fig. 15.1). A portfolio—as a whole—forms a deterministic system, the parts of which possess options; that is why it considered it a macrodeterministic system, which also has a degree of freedom [6]. There is a pressing need for sustainable, yet growthgenerating, reliable, and effective theories and practices in the financial field. My model, created from the synthesis of chaos theory and system theory, is an initial condition for portfolio management that strives to include entropy, i.e. order into finance [7]. This study presents a model that epitomizes such a formal set of abstract systems, which according to Mesarovic’s thoughts [8], simulates the properties of this system with simple mathematical structures and, through this, finds a common set of special theories. The model offers such a heuristic pattern of diversification that aims to support Thaler’s behavioural finance [9] by defining the optimum distribution in 1:1:1:4 based on the entropy of nature (the principle of similarity of triangles). Namely, restricted human rationality and the behavioural forms of finance are barriers that can arrest rational behaviour. Such decisions, therefore, require simplified models that stand as a theoretical and practical example through the M&M model presented in this study. Since such constructions are proven solely and precisely by their functioning [10] the study also shows the actual, decade, and crisis yields, which demonstrates the usefulness of importing nature distribution into portfolio management. This demonstrates the conjecture in system research completed in the 1970s. More specifically, that the structure of science is isomorphic with the structure of nature [11]. “Each system consists of local and global components”—says Ludwig von Bertalanffy, the Father of General System Theory. Based on this, the examined 4-element portfolios are also divided into local and global components (Table 15.1). In the research I use a heuristic decision making method (1 trade annually). In addition, the self-developed interdisciplinary model also serves as a basis for a process called Rolling Nuts (Fig. 15.2), which signifies a continuous value creation principle. Fig. 15.1 The M&M model and its distribution

15 Chaotic Solutions for Asset Management Complexity Table 15.1 The research used investment classes

161

Investment class

In research

Potential

Stock

DAX

local

Bond

HU

local

Gold

GOLD

global

Commodities

NSDQ

global

Fig. 15.2 The Rolling Nuts Portfolios, focused on different objects

The principle provides for the reorganization and conversion of portfolios according to the model, which manages the portfolio in a dynamic way, i.e. contributes to the control and leadership of a reactive financial system. The current low interest rate environment has created a growing demand for constructions that are both secure and profitable, and responsive to opportunities/threats. My study demonstrates how the Mandelbrot set becomes measurable and how it becomes a base for a self-sustained and self-stimulated financial portfolio because of its system properties. In this case, portfolio growth is not due to the state of equilibrium, but its absence and inaccessibility, as declared by Prigogine [12]. My study proves that portfolios are examples of such open, dynamic systems; that is, free in their decisions, but determined in their frameworks (complexity). In this case, the model I synthesized in an interdisciplinary manner from chaos and system theory serves a reference point, not an equilibrium point. I have supported and proved this theory with concrete, practical examples, which make the reader aware of the equilibrium (optimal) state discovered. The strategy is built around this, and it can be safely constructed in both the short and the long term, or even in times of crisis. The Rolling Nuts Portfolio based on a passive investment strategy is compared—with diversification perspective—to Brown’s [13] Permanent Portfolio (1:1:1:1), with the balanced 1:1:1:2 ratio, and the 1:1:1:3 portfolio distributions, and yielding the ‘classic’ 60/40 benchmark portfolio (Fig. 15.3), to get relevant results and conclusions. In addition to return, the study specifically discusses the transformability of the model, enabling the strategy to be active and flexible by responding to changes in the environment, but respecting the original initial conditions and the limited range.

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Fig. 15.3 The analyzed portfolio distributions

On Fig. 15.4 can be seen the difference between the yields of the 4 Rolling Nuts (RN) portfolios and the other distributions, over the past 20 years. The results show the power of pure diversification. The best result was achieved by the RN (Alt.) Portfolio, with a 549% increase in assets. The second best was the Permanent Portfolio, with 492%, and the third the RN (Com.) Portfolio with 482%. This proves that the distribution based on the entropy of nature can be successful as a portfolio and also proves that a permanent distribution is a viable long-term savings/profitability method. The worst result of the analysis brings the classic 60/40 portfolio, with only 356% growth. This is about 200% less than the best result. This proves that the classic portfolio distribution is outdated, it is not worthwhile to divide the investments into two, because nowadays there are many more financial products available, which

560% 482%

480% 400%

356

320% 240% 160% 80% RN (Alt.)

RN (Com.)

RN (Stock)

RN (Gua.)

PP

40/20

1/2

60/40

Fig. 15.4 20-year portfolio comparison

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Fig. 15.5 The lifecycle portfolios, in years

we can choose from. The so far analyzed portfolios are, however, only simple distributions, without management methods. Nevertheless, portfolio management can modify the performance, with portfolio restructuring and life cycle management. The M&M Model permits to build up various focuses and lifecycles. I created 3 different lifecycles in this 20-year investment period, what reorganize portfolios in different intervals. Always based on the distribution of the M&M model, they only change the focus. The Lifecycle I. Portfolio reorganizes the distribution every 2 years, so making 10 rearrangements over this period. The Lifecycle II. Portfolio is restructured every 5 years, which means 4 rearrangements, while Lifecycle III. Portfolio rebalances only every 7 years (Fig. 15.5). “A system is a set of processes, that become visible as temporary structures. The organization is constantly evolving: it shuts off the old structure and builds new ones when it becomes necessary” [14]. My research focused on—with creating Lifecycle Portfolios—to improve performance with management tools. In order to compare the results, I compared the best 3 portfolios (from Fig. 15.4) with the 3 life cycle managed portfolios. The results are shown in Fig. 15.6. All three Lifecycle Portfolios outperformed the results so far, in order: Lifecycle III. (891%), Lifecycle II. (874%) and Lifecycle I. (795%). This proves the potential of reorganization, namely the difference between the best and worst results in Fig. 15.6 is 400%. The Table 15.2 summarizes Rolling Nuts-based Portfolio Growth, Average annual yield, Volatility and—as always relevant in investment field—the Returns. The annual yield column shows the average of the yields achieved in each year, based on these the Lifecycle II. Portfolio proved to be the best investment with a 12.53% annual average return. The two lower rows of Table 15.2 also shows the averages. From this, it can be concluded, that simple on Rolling Nuts basis diversified portfolios,

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900%

795%

700% 500% 300% 100% RN (Alt.) RN Lifecycle I.

RN (Com.) RN Lifecycle II.

PP RN Lifecycle III.

Fig. 15.6 The power of restructuring

Table 15.2 20 years rolling nuts results

1999–2018 RN (Com.) RN (Stock) RN (Alternative) RN (Guaranteed) RN Lifecycle I. RN Lifecycle II. RN Lifecycle III. RN Average RN Lifecycle av.

Growth (%) 482 390 549 459 795 874 891 470 853

Av. an. yield (%) 11.11 9.39 9.70 8.40 11.67 12.53 12.29 9.65 12.16

Volatility 114.99 85.36 44.82 42.93 47.63 65.94 48.45 72.03 54.01

Return (years) 12 12 8 8 7 7 8 10 7

without any restructuring, would have produced an average yield of 9.65% over the past 20 years. For Lifecycle Portfolios is this value 12.16%. I think any of us would be happy with such a passive source of income.

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15.3 Conclusion The research results confirm there can be a diversification policy that is both costeffective and secure. I can prove this through the example of the Rolling Nuts Portfolio implemented in practice, which—counter to the literature—can precisely define the initial condition of chaos theory. With this introduced system theory application in the portfolio management field, I seek to provide a traceable example of a lucrative and sustainable model, one that helps achieve secure and profitable returns for both private and institutional investors. Acknowledgements This paper was made in frame of the EFOP-3.6.1-16-2016-00018—Improving the role of research + development + innovation in the higher education through institutional developments assisting intelligent specialization in Sopron and Szombathely.

References 1. W.R. Ashby, Introductory remarks at a panel discussion, in Views on General Systems Theory (Wiley, New York, 1964), pp. 165–169 2. H. Mintzberg, The Structuring of Organizations: A Synthesis of the Research (Prentice-Hall, Englewood Cliffs, NJ, 1979). https://doi.org/10.1007/978-1-349-20317-8_23 3. D.F. Swensen, Pioneering Portfolio Management (Free Press, 2009). ISBN 1416544690 4. R.G. Hagstrom, The Warren Buffett Portfolio: Mastering the Power of the Focus Investment Strategy (Wiley, 2001) 5. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Co, 1982). https://doi. org/10.1119/1.13295 6. E. Laszlo, Evolution: The General Theory (Hampton Press, New Jersey, 1996) 7. G. Cziraki, Die Universalität von Anlagestrategien. E-CONOM VII./1., 75–86 (2018) 8. M. Mesarovic, On some metamathematical results as properties of general systems. Math Syst Theory 2(N4), 357–361 (1968) 9. S. Benartzi, R.H. Thaler, Myopic loss aversion and the equity premium puzzle. Q. J. Econ. 110(1), 73–92 (1995) 10. J. Neumann, The Mathematician. Work Mind 1(1), 180–196 (1947) 11. V.N. Sadovsky, Foundations of General Systems Theory (Izd. Nauka, Moskow, 1974) 12. I. Prigogine, La nouvelle alliance. Métamorphose de la science (Gallimard, Paris, 1986) 13. H. Brown, Fail-Safe Investing: Lifelong Financial Security in 30 Minutes (St. Martin’s Press, 1999) 14. M. Wheatley, Leadership and the New Science: Discovering Order in a Chaotic World (BerrettKoehler Publishers, Inc., 2000)