In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear dynamical sy

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*English*
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- Siyuan Xing
- Albert C J Luo

*Table of contents : PrefaceIntroductionA Semi-Analytical MethodPeriodic Motions in Time-Delay Systems Discretization Formulation for Period-m MotionsA Global Sequential ScenarioFrequency-Amplitude Analysis Finite Fourier Series Frequency-Amplitude Characteristics Asymmetric Period-1 to Period-8 Motions Asymmetric Period-2 to Period-8 Motions Asymmetric Period-3 to Period-6 Motions Symmetric Period-3 to Period-6 Motions Symmetric Period-5 to Period-10 MotionsGlobal Sequential Periodic Motions A Symmetric Period-1 Motion Asymmetric Period-1 to Period-8 Motions Symmetric Period-3 to Period-6 Motions Asymmetric Period-2 to Period-8 Motions Symmetric Period-5 to Period-10 Motions Asymmetric Period-3 to Period-6 MotionsConclusive RemarksReferencesAuthors' BiographiesBlank Page*

Series ISSN: 2573-3168

XING • LUO

Synthesis Lectures on Mechanical Engineering Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems

In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is presented for a better understanding of global behaviors and motion transitions for one periodic motion to another one. A 1-dimensional (1-D), time-delayed, nonlinear dynamical system is considered as an example to show how to determine the global sequential scenarios of the bifurcation trees of periodic motions to chaos. All stable and unstable periodic motions on the bifurcation trees can be determined. Especially, the unstable periodic motions on the bifurcation trees cannot be achieved from the traditional analytical methods, and such unstable periodic motions and chaos can be obtained through a specific control strategy. The sequential periodic motions in such a 1-D time-delayed system are achieved semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. Each bifurcation tree of a specific periodic motion to chaos are presented in detail. The bifurcation tree appearance and vanishing are determined by the saddle-node bifurcation, and the cascaded period-doubled periodic solutions are determined by the period-doubling bifurcation. From finite Fourier series, harmonic amplitude and harmonic phases for periodic motions on the global bifurcation tree are obtained for frequency analysis. Numerical illustrations of periodic motions are given for complex periodic motions in global bifurcation trees. The rich dynamics of the 1-D, delayed, nonlinear dynamical system is presented. Such global sequential periodic motions to chaos exist in nonlinear dynamical systems. The frequency-amplitude analysis can be used for re-construction of analytical expression of periodic motions, which can be used for motion control in dynamical systems

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SEQUENTIAL BIFURCATION TREES TO CHAOS IN NONLINEAR TIME-DELAY SYSTEMS

Siyuan Xing, California Polytechnic State University Albert C.J. Luo, Southern Illinois University

Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems Siyuan Xing Albert C.J. Luo

Synthesis Lectures on Mechanical Engineering

Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems

Synthesis Lectures on Mechanical Engineering Synthesis Lectures on Mechanical Engineering series publishes 60–150 page publications pertaining to this diverse discipline of mechanical engineering. The series presents Lectures written for an audience of researchers, industry engineers, undergraduate and graduate students. Additional Synthesis series will be developed covering key areas within mechanical engineering.

Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems Siyuan Xing and Albert C.J. Luo 2020

Introduction to Deep Learning for Engineers: Using Python and Google Cloud Platform Tariq M. Arif 2020

Towards Analytical Chaotic Evolutions in Brusselators Albert C.J. Luo and Siyu Guo 2020

Modeling and Simulation of Nanoﬂuid Flow Problems Snehashi Chakraverty and Uddhaba Biswal 2020

Modeling and Simulation of Mechatronic Systems using Simscape Shuvra Das 2020

Automatic Flight Control Systems Mohammad Sadraey 2020

Bifurcation Dynamics of a Damped Parametric Pendulum Yu Guo and Albert C.J. Luo 2019

iv

Reliability-Based Mechanical Design, Volume 2: Component under Cyclic Load and Dimension Design with Required Reliability Xiaobin Le 2019

Reliability-Based Mechanical Design, Volume 1: Component under Static Load Xiaobin Le 2019

Solving Practical Engineering Mechanics Problems: Advanced Kinetics Sayavur I. Bakhtiyarov 2019

Natural Corrosion Inhibitors Shima Ghanavati Nasab, Mehdi Javaheran Yazd, Abolfazl Semnani, Homa Kahkesh, Navid Rabiee, Mohammad Rabiee, and Mojtaba Bagherzadeh 2019

Fractional Calculus with its Applications in Engineering and Technology Yi Yang and Haiyan Henry Zhang 2019

Essential Engineering Thermodynamics: A Student’s Guide Yumin Zhang 2018

Engineering Dynamics Cho W.S. To 2018

Solving Practical Engineering Problems in Engineering Mechanics: Dynamics Sayavur Bakhtiyarov 2018

Solving Practical Engineering Mechanics Problems: Kinematics Sayavur I. Bakhtiyarov 2018

C Programming and Numerical Analysis: An Introduction Seiichi Nomura 2018

Mathematical Magnetohydrodynamics Nikolas Xiros 2018

v

Design Engineering Journey Ramana M. Pidaparti 2018

Introduction to Kinematics and Dynamics of Machinery Cho W. S. To 2017

Microcontroller Education: Do it Yourself, Reinvent the Wheel, Code to Learn Dimosthenis E. Bolanakis 2017

Solving Practical Engineering Mechanics Problems: Statics Sayavur I. Bakhtiyarov 2017

Unmanned Aircraft Design: A Review of Fundamentals Mohammad Sadraey 2017

Introduction to Refrigeration and Air Conditioning Systems: Theory and Applications Allan Kirkpatrick 2017

Resistance Spot Welding: Fundamentals and Applications for the Automotive Industry Menachem Kimchi and David H. Phillips 2017

MEMS Barometers Toward Vertical Position Detection: Background Theory, System Prototyping, and Measurement Analysis Dimosthenis E. Bolanakis 2017

Engineering Finite Element Analysis Ramana M. Pidaparti 2017

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All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations in printed reviews, without the prior permission of the publisher. Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems Siyuan Xing and Albert C.J. Luo www.morganclaypool.com ISBN: 9781681739427 ISBN: 9781681739434 ISBN: 9781681739441

paperback ebook hardcover

DOI 10.2200/S01038ED1V01Y202008MEC031

A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MECHANICAL ENGINEERING Lecture #31 Series ISSN Print 2573-3168

Electronic 2573-3176

Sequential Bifurcation Trees to Chaos in Nonlinear Time-Delay Systems

Siyuan Xing California Polytechnic State University, San Luis Obispo, CA

Albert C.J. Luo Southern Illinois University, Edwardsville, IL

SYNTHESIS LECTURES ON MECHANICAL ENGINEERING #31

M &C

Morgan

& cLaypool publishers

ABSTRACT In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos in nonlinear dynamical systems is presented for a better understanding of global behaviors and motion transitions for one periodic motion to another one. A 1-dimensional (1-D), time-delayed, nonlinear dynamical system is considered as an example to show how to determine the global sequential scenarios of the bifurcation trees of periodic motions to chaos. All stable and unstable periodic motions on the bifurcation trees can be determined. Especially, the unstable periodic motions on the bifurcation trees cannot be achieved from the traditional analytical methods, and such unstable periodic motions and chaos can be obtained through a speciﬁc control strategy. The sequential periodic motions in such a 1-D time-delayed system are achieved semianalytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. Each bifurcation tree of a speciﬁc periodic motion to chaos are presented in detail. The bifurcation tree appearance and vanishing are determined by the saddle-node bifurcation, and the cascaded period-doubled periodic solutions are determined by the period-doubling bifurcation. From ﬁnite Fourier series, harmonic amplitude and harmonic phases for periodic motions on the global bifurcation tree are obtained for frequency analysis. Numerical illustrations of periodic motions are given for complex periodic motions in global bifurcation trees. The rich dynamics of the 1-D, delayed, nonlinear dynamical system is presented. Such global sequential periodic motions to chaos exist in nonlinear dynamical systems. The frequency-amplitude analysis can be used for re-construction of analytical expression of periodic motions, which can be used for motion control in dynamical systems.

KEYWORDS 1-dimensional time-delayed system, global sequential scenario of bifurcation trees, implicit mapping, mapping structures, nonlinear frequency-amplitudes

ix

xi

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2

A Semi-Analytical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3

Periodic Motions in Time-Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Formulation for Period-m Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4

A Global Sequential Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5

Frequency-Amplitude Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.1 Finite Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Frequency-Amplitude Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2.1 Asymmetric Period-1 to Period-8 Motions . . . . . . . . . . . . . . . . . . . . . 34 5.2.2 Asymmetric Period-2 to Period-8 Motions . . . . . . . . . . . . . . . . . . . . . 35 5.2.3 Asymmetric Period-3 to Period-6 Motions . . . . . . . . . . . . . . . . . . . . . 35 5.2.4 Symmetric Period-3 to Period-6 Motions . . . . . . . . . . . . . . . . . . . . . . 40 5.2.5 Symmetric Period-5 to Period-10 Motions . . . . . . . . . . . . . . . . . . . . . 45

6

Global Sequential Periodic Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.1 A Symmetric Period-1 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.2 Asymmetric Period-1 to Period-8 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.3 Symmetric Period-3 to Period-6 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.4 Asymmetric Period-2 to Period-8 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.5 Symmetric Period-5 to Period-10 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.6 Asymmetric Period-3 to Period-6 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7

Conclusive Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

xiii

Preface Periodic motions to chaos in nonlinear dynamical systems have been of great interest recently. The global sequential orders of periodic motions to chaos in nonlinear dynamical systems are important for a better understanding of global nonlinear dynamics of a dynamical system, and the relation between two bifurcation trees of periodic motions to chaos can be determined. Herein, the global sequential orders of bifurcation trees of periodic motions to chaos are discussed through a 1-dimensional (1-D), ﬁrst-order, nonlinear, time-delay system. The semianalytical method and frequency-amplitude analysis of bifurcation trees of periodic motions to chaos are presented herein. In this book, a brief introduction of periodic motions in nonlinear time-delay systems is given in Chapter 1. In Chapter 2, the semi-analytical method for periodic motions in timedelay systems is presented, which will be used in this book. To present a global sequential order of periodic motions to chaos in time-delay dynamical systems, a 1-D, ﬁrst-order, time-delay, nonlinear system is discussed as an example in Chapter 3, which is one of simplest time-delay systems. Such a system is an Ikeda-like system and can be implemented in electronic circuit. In Chapter 4, the periodic motions and bifurcations in a 1-D time-delayed, nonlinear systems are given semi-analytically, and a sequential order of bifurcation trees of periodic motions to chaos are given for a better understanding of motion complexity in lowest dimensional systems. From ﬁnite Fourier series, harmonic frequency-amplitude characteristics of periodic motions in the bifurcation trees are presented in Chapter 5. In Chapter 6, numerical illustrations of periodic motions are given from analytical predictions. Phase trajectories, harmonic amplitudes, and phases of periodic motions are illustrated. The rich dynamics of the 1-D, time-delay, nonlinear dynamical system is presented. Finally, the authors hope the materials presented herein can give some inspiration on the global dynamics of nonlinear dynamical systems. Through such research, one can control unstable periodic motion and unstable chaos in nonlinear dynamical systems through timedelay. Siyuan Xing and Albert C.J. Luo September 2020

1

CHAPTER

1

Introduction One has been interested in periodic motions for a long time. For instance, in 1788, Lagrange [1] developed the method of averaging for periodic motions of three-body problem. To continue Langrange’s work, in 1890, Poincare [2] developed the perturbation theory for periodic motions of celestial bodies. In 1920, van der Pol [3] applied the method of averaging for the periodic solutions of oscillation systems in circuits. In 1928, Fatou [4] gave the proof of the asymptotic validity through the solution existence theorems of diﬀerential equations. In 1935, Krylov and Bogoliubov [5] further developed the method of averaging for nonlinear oscillations in nonlinear vibration systems. Since then, researchers extensively used the perturbation method to investigate periodic solutions in nonlinear dynamical systems. In 1964, Hayashi [6] discussed perturbation methods, averaging method, and principle of harmonic balance. The stability of approximate periodic solutions in nonlinear dynamical systems is determined by the improved Mathieu equation. In 1973, Nayfeh [7] systematically presented the perturbation theory and multi-scale methods. In 1976, Holmes and Rand [8] used the method of averaging to discuss the stability and bifurcation of the Duﬃng oscillator through the catastrophe theory. The catastrophe theory was used to determine the singular characteristics of frequency-amplitude curves. In 1979, Nayfeh and Mook [9] used the multiple-scale perturbation methods for approximate solutions of periodic motions. The frequency-amplitude responses were discussed. In 1987, Rand and Armbruster [10] presented the perturbation method and bifurcation theory to determine the stability of periodic solutions. To determine frequency-amplitude characteristics of periodic motions in nonlinear dynamical systems, the harmonic balance method is often used as one of the most popular techniques. In 1997, Luo and Han [11] analytically studied the stability and bifurcations of periodic solutions of Duﬃng oscillators through the ﬁrst-order harmonic balance method. To obtain accurate results of periodic solutions in nonlinear vibration, many harmonic terms are included in the harmonic balance method. In 2008, Peng et al. [12] presented the approximate period-1 solution for the Duﬃng oscillator by the HB3 method compared with the fourth-order Runge–Kutta method. Such results still cannot give a satisfactory solution. Time-delay, nonlinear dynamical systems are of great interest in physiology, optics, and engineering in recent years. Mackey and Glass [13] characterized the physiological diseases by a 1-D, time-delayed dynamical system and related the onset of diseases to the bifurcations of the dynamical system. Ikeda et al. [14, 15] modeled an optical bistable resonator using a delaydiﬀerential equation and predicted a memory device by using the multi-stable periodic motions in such a system. Based on Ikeda’s work, Larger et al. [16] developed an optoelectronics model

2

1. INTRODUCTION

to implement the chaos-based encryption demonstrators. Yang et al. [17] studied the vibration resonance in a Duﬃng system with generalized delay feedback in a fractional-order diﬀerential form. Jeevarathinam et al. [18] further investigated the Duﬃng systems with a gamma distributed time-delayed feedback and an integrative time-delayed feedback. Time-delay systems are also studied for synchronization. For instance, Krishnaveni and Sathiyanathan [19] investigated the synchronization of couple map lattice using delayed variable feedback. Akhmet [20] studied the synchronization of the cardiac pacemaker model with delayed pulse-coupling. Bernstein and Rand [21] investigated the delay-coupled Mathieu equations in synchrotron dynamics. Time-delay feedbacks in the slow ﬂow for the time-delayed nonlinear dynamical system were discussed in Sah and Rand [22]. Shahverdiev et al. [23] analyzed the synchronization of two coupled semiconductor lasers with time delays. Recently, chaotic motions in dynamical systems with simple algebra form were of great interest. 1-D, time-delay systems were studied for global dynamics of periodic motion to chaos. Ucar [24] proposed a 1-D delay-diﬀerential system with cubic-nonlinearity delay feedback. Sprott [25] presented a simple 1-D time-delay system with sinusoidal nonlinearity. Such 1D, time-delay systems can be easily implemented through electrical circuits. Namajūnas [26] designed an analog circuit with a tuning delay component and demonstrated the existence of chaotic motions in such a system. Voss [27] anticipated chaotic signals in a 1-D, delaydiﬀerential system using an electronic circuit. To understand the complex phenomena in nonlinear systems, periodic motions are very important. Numerical methods were applied to study the periodic motions extensively. Engelborghs et al. [28] applied collection method to periodic motions in autonomous delaydiﬀerential equations. Insperger and Stepan [29, 30] developed the semi-discretization method for stability analysis of linear periodic time-delay systems. To analytically predict periodic motions in delayed, nonlinear dynamical systems, one employed the perturbation method (e.g., [31], [32]) and the harmonic balance method (e.g., [33], [34]). Such methods are adequate only for simple periodic motions, but not suitable for complex periodic motions. Thus, in 2012, Luo [35] systematically developed a generalized harmonic balance method for analytical solutions of periodic motions in nonlinear dynamical systems with a speciﬁc computational accuracy. This method is to transform the time-domain responses to the frequency-domain responses in nonlinear dynamical systems. However, the frequency domain responses varying with time are treated as a dynamical system from which the stability and bifurcation of frequency-domain responses can be determined, and the stability and bifurcation of time-domain responses in nonlinear dynamical systems can be determined through the frequency domains. In fact, such a method is also a generalized method of averaging for all harmonic terms rather than the constant term only in the traditional method of averaging. Luo and Huang [36] applied the generalized harmonic method to the Duﬃng oscillator for approximate solutions of periodic motions, and Luo and Huang [37] presented the analytical bifurcation trees of period-m motions to chaos in the Duﬃng oscillator (see also Luo and Huang [38, 39]). Luo [40] extended the generalized

1. INTRODUCTION

harmonic balanced method for delayed, nonlinear systems. Luo and Jin [41] used such a generalized harmonic balance method for the time-delayed, quadratic nonlinear oscillator. Luo and Jin [42, 43] analytically obtained the bifurcation trees of periodic motion in a Duﬃng oscillator with time-delay. The generalized harmonic balance method is diﬃcult to be applied for periodic motions in dynamical systems with non-polynomial nonlinearity. Hence, Luo [44] developed a semianalytical method to determine periodic motions in nonlinear systems with/without delay. Such an approach develops discrete implicit mappings for periodic motions with controllable accuracy. Luo and Guo [45] applied such a method for periodic motions in a nonlinear Duﬃng oscillator and compared to the generalized harmonic balance method. Luo and Xing [46] studied bifurcation trees of period-1 motions to chaos in a delayed, hardening Duﬃng oscillator through the semi-analytical method. Luo and Xing [47] also investigated periodic motions in a time-delayed, twin-well, Duﬃng oscillator and discussed the possibility of inﬁnity bifurcation trees in such a nonlinear oscillator. Luo and Xing [48] also investigated the time-delay eﬀects on periodic motions in a time-delayed, Duﬃng oscillator. Luo and Xu [49] studied a twodegree-of-freedom van der Pol-Duﬃng oscillator and discovered a series of periodic motions. A few studies for periodic motions in 1-D time-delayed, nonlinear systems was completed but no signiﬁcant results were obtained. Xing and Luo [50] presented the global sequential order of periodic motions to chaos in a ﬁrst-order, nonlinear time-delay system. Herein, the main results will be from that paper. In this book, a brief introduction of periodic motions in nonlinear time-delay systems is given in Chapter 1. In Chapter 2, the semi-analytical method for periodic motions in timedelay systems will be presented, which will be used in this book. To present a global sequential order of periodic motions to chaos in time-delay dynamical systems, a 1-D, ﬁrst-order, timedelay, nonlinear system will be presented in Chapter 3, which is one of simplest time-delay systems. Such a system is an Ikeda-like system and can be implemented in electronic circuit. In Chapter 4, the periodic motions and bifurcations in a 1-D time-delayed, nonlinear systems will be given, and a series of bifurcation trees of periodic motions to chaos will be presented for a better understanding of motion complexity in lowest dimensional nonlinear systems. From ﬁnite Fourier series, harmonic frequency-amplitude characteristics of periodic motions in the bifurcation trees will be analyzed in Chapter 5. In Chapter 6, numerical illustrations of periodic motions in such a 1-D time-delay nonlinear system will be given from analytical predictions. Phase trajectories, harmonic amplitudes and phases of periodic motions will be demonstrated. The rich dynamics of the 1-D time-delayed, nonlinear dynamical system will be presented.

3

5

CHAPTER

2

A Semi-Analytical Method From Luo [44], a period-m ﬂow in a time-delayed, nonlinear dynamical system can be described through discrete nodes for period-mT . To determine period-m motion in the time-delay dynamical systems, the following theorem is presented herein. Theorem 2.1 [44].

Consider a time-delay nonlinear dynamical system xP D f .x; x ; t; p/ 2 Rn ;

with x.t0 / D x0 ; x.t/ D ˆ .x0 ; t

t0 ; p/ for t 2 Œt0

; 1/:

(2.1)

If such a time-delay dynamical system has a period-m ﬂow x.m/ .t/ with ﬁnite norm jjx.m/ jj and period mT (T D 2=), there is a set of discrete time tk (k D 0; 1; : : : ; mN ) with (N ! 1) during mperiods (mT ), and the corresponding solution x.m/ .tk / and vector ﬁelds f.x.m/ .tk /; x.m/ .tk /; tk ; p/ and xk.m/ are on the approximate solution of the periodic ﬂow are exact. Suppose discrete nodes x.m/ k jj "k and jjx .m/ .tk / xk.m/ jj "k with small "k ; "k 0 and under jjx.m/ .tk / x.m/ k

.m/

.m/ .m/ (2.2)

f.x .tk /; x .m/ .tk /; tk ; p/ f.xk ; xk tk ; p/ ık with a small ık 0. During a time interval t 2 Œtk ; x.m/ / (k D 1; 2; : : : ; mN ) as .x.m/ k k

1 ; t k ,

; x.m/ / ! there is a mapping Pk W .x.m/ k 1 k 1

; x.m/ ; p/ D 0; ; x.m/ I x.m/ / D Pk .x.m/ ; x.m/ / with gk .x.m/ ; x.m/ .x.m/ k k 1 k k 1 k k k 1 k 1 .m/ xj.m/ D hj .x.m/ rj 1 ; xrj ; rj /; j D k; k .m/ .e.g., xr .m/ D x.m/ sr C r .xrr 1

1I rj D j

x.m/ rr /; r D

1 Œ hrj

lj ; k D 1; 2; : : : ; mN I l

rj X

i D1

(2.3)

hrj Ci /:

where gk is an implicit vector function and hj is an interpolation vector function. Consider a mapping structure as P D PmN ı PmN

1

ı ı P2 ı P1 W x.m/ ! x.m/ 0 mN I with

Pk W .x.m/ ; x.m/ / ! .x.m/ ; x.m/ / .k D 1; 2; : : : ; mN /: k 1 k 1 k k

(2.4)

6

2. A SEMI-ANALYTICAL METHOD .m/ .m/ ; xk.m/ / (k D 0; 1; : : : ; mN ) computed For x.m/ /, if there is a set of points .x.m/ mN D P .x0 ; x0 k by

)

gk .x.m/ ; x.m/ I x.m/ ; x.m/ ; p/ D 0; k 1 k k 1 k

.m/ ; rj /; j D k; k xj.m/ D hj .x.m/ rj 1 ; xrj .m/ x.m/ rj 1 D xmod.rj

1CmN;mN /

.k D 1; 2; : : : ; mN /

1

(2.5)

; x.m/ D x.m/ I rj mod.rj CmN;mN /

.m/ .m/ x.m/ D x.m/ D xmN : 0 mN and x0

Then the points x.m/ and xk.m/ (k D 0; 1; : : : ; mN ) are the approximation of points x.m/ .tk / and k x.m/ .tk / of periodic solutions. In the neighborhoods of x.m/ and x.m/ , with x.m/ D x.m/ C x.m/ k k k k k .m/ .m/ and x.m/ D x C x , the linearized equation is given by k k k k X j Dk 1

@gk

xj.m/ C .m/

@xj

with rj D j

@xj.m/ .m/ @xj.m/ . xrj C .m/ x.m/ rj 1 / D 0 .m/ .m/ @xj @xrj @xrj 1 @gk

lj ; j D k

1; kI .k D 1; 2; : : : ; mN /:

The resultant Jacobian matrices of the periodic ﬂow are " .m/ # @yk DPk.k 1/:::1 D D Ak Ak 1 : : : A1 .k D 1; 2; : : : ; mN /; .m/ .m/ @y.m/ 0 .y0 ;:::;yk / " .m/ # @ymN D AmN AmN 1 : : : A1 ; and DP D DPmN.mN 1/:::1 D .m/ .m/ @y.m/ 0 .y ;:::;y / 0

where y.m/ k

D

(2.6)

A.m/ y.m/ ; k 1 k

A.m/ k

(2.7)

mN

"

@y.m/ k

#

@y.m/ k 1

(2.8)

.m/ .m/ .yk 1 ;yk /

and "

B.m/ k

A.m/ k

D

B.m/ k

h D .a.m/ k.k

I.m/ k 0.m/ k

.a.m/ k.rk

I.m/ k

/ 1/ nn .m/ 0k 1

# n.sC1/n.sC1/

/ ; 0 ; : : : ; .a.m/ / 1/ nn nn k.rk 1/ nn

D diag .Inn ; Inn ; : : : ; Inn /nsns ; D .0nn ; 0nn ; : : : ; 0nn /T I ƒ‚ … „ s

i

;

; s D 1 C lk

1

(2.9)

2. A SEMI-ANALYTICAL METHOD

y.m/ k .m/ yk 1 y.m/ k .m/ yk 1

a.m/ kj a.m/ krj a.m/ k.rj 1/

D D

T .x.m/ ; x.m/ ; : : : ; x.m/ rk 1 / ; k k 1 T .x.m/ ; x.m/ ; : : : ; x.m/ rk 1 1 / ; k 1 k 2

T D .x.m/ ; x.m/ ; : : : ; x.m/ rk 1 / ; k k 1

D .x.m/ ; x.m/ ; : : : ; x.m/ rk 1 k 1 k 2

D D D

with rj D j

" " "

@gk @x.m/ k @gk @x.m/ k @gk

# # #

1

@gk @xj.m/

1 j C1 X ˛Dj 1

@x.m/ k lj ; j D k

1/

(2.10) T

I

;

@gk @x.m/ ˛ ; .m/ @x rj @x˛

j X ˛Dj 1

(2.11)

@gk @x.m/ ˛ .m/ @x rj 1 @x˛

1; k:

(k D 1; 2; : : : ; mN ) can be estimated by the eigenvalues of The properties of discrete points x.m/ k DPk.k 1/:::1 as ˇ ˇ N n.sC1/n.sC1/ ˇ D 0 .k D 1; 2; : : : ; mN /: ˇDPk.k 1/:::1 I (2.12) The eigenvalues of DP for such periodic motion are determined by ˇ ˇ ˇDP In.sC1/n.sC1/ ˇ D 0;

(2.13)

and the stability and bifurcation of the periodic ﬂow can be classiﬁed by the eigenvalues of DP.y0 / with o m o (2.14) Œnm 1 ; n1 W Œn2 ; n2 W Œn3 ; 3 W Œn4 ; 4 jn5 W n6 W Œn7 ; l; 7 : (i) If the magnitudes of all eigenvalues of DP are less than one (i.e., ji j < 1, i D 1; 2; : : : ; n.s C 1/), the approximate periodic solution is stable. (ii) If at least the magnitude of one eigenvalue of DP is greater than one (i.e., ji j > 1, i 2 f1; 2; : : : ; n.s C 1/g), the approximate periodic solution is unstable. (iii) The boundaries between stable and unstable periodic ﬂow with higher-order singularity give bifurcation and stability conditions with higher-order singularity. Proof. See Luo [44].

7

9

CHAPTER

3

Periodic Motions in Time-Delay Systems In this chapter, formulations for periodic motions in a simple time-delayed nonlinear system is presented. The diﬀerential equation of such a time-delay system is discretized ﬁrst for discrete mappings with prescribed accuracy, and the corresponding periodic motions are determined through speciﬁc mappings structures. The stability and bifurcations of periodic motions in such a time-delay nonlinear system are determined by eigenvalue analysis.

3.1

DISCRETIZATION

Consider a 1-D, time-delay, nonlinear system as xP D ˛1 x

˛2 sin x

ˇx 3 C Q0 cos t;

(3.1)

where x D x.t / and x D x.t /. ˛1 ; ˛2 ; ˇ are system coeﬃcients; and Q0 are excitation frequency and excitation amplitude, respectively. The system without the cubic polynomial is an Ikeda system. The term of ˇx 3 is added to impose a strong constraint for the motions far from the origin. The delay herein is corresponding to the propagation delay. Such a system can be implemented in an electronics circuit. Let xk D x.tk / at tk D kh .k D 0; 1; 2 : : :/ be the sampled nodes with uniform time-step on a periodic motion in the 1-D, time-delayed, nonlinear dynamical system. xk D x.tk / is the corresponding delayed point of xk . By the mid-point discretization scheme for the nonlinear dynamical system, an implicit map Pk from .xk 1 ; xk 1 / to .xk ; xk / for the interval of t 2 Œtk 1 ; tk is constructed as Pk W xk

1 ; xk 1

! xk ; xk

) xk ; xk D Pk xk

1 ; xk 1

:

(3.2)

The mapping Pk is expressed by xk

xk

1

1 1 D hf ˛1 .xk C xk 1 / ˛2 sinŒ xk C xk 1 2 2 1 1 3 ˇ .xk C x k 1 / C Q0 cos .t C h/g: 8 2

(3.3)

10

3. PERIODIC MOTIONS IN TIME-DELAY SYSTEMS

The time-delay node xj .j D k nodes xrj 1 and xrj as xj D hj xrj

1; k/ is interpolated by two adjacent non-delay-node

1 ; xrj ; rj

for rj D j

(3.4)

lj ;

where lk D int.= h/. Using a simple Lagrange interpolation, the time-delay discrete node xj D hj .xrj 1 ; xrj ; rj / (j D k; k 1) is represented by xk D xk

lk 1

C .1

1 C lk / xk h

xk

lk

lk 1

:

(3.5)

Now the discretization of the delayed nonlinear dynamical system is performed.

3.2

FORMULATION FOR PERIOD-m MOTIONS

In the 1-D, time-delay, nonlinear dynamical system, the mapping structure of a period-m motion is .m/ .m/ ; xmN /; P D PmN ı PmN 1 ı ı P2 ı P1 W .x0.m/ ; x0.m/ / ! .xmN ƒ‚ … „

(3.6)

/ ! .xk.m/ ; xk.m/ / Pk W .xk.m/1 ; xk.m/ 1

(3.7)

mN actions .m/ .m/ .xmN ; xmN / D P .x0.m/ ; x0 .m/ /;

with

.k D 1; 2; : : : ; mN /:

The nodes xk.m/ .k D 1; 2; : : : ; mN ) on the period-m motion of the 1-D, time-delayed nonlinear system are predicted by )

; xk.m/ ; p/ D 0 ; xk.m/ ; xk .m/ gk .xk.m/ 1 1

xj .m/ D hj .xr.m/ ; xr.m/ ; rj /; j D k; k j j 1

1

.k D 1; 2; : : : ; mN /:

(3.8)

.m/ .m/ x0.m/ D xmN and x0.m/ D xmN

The corresponding algebraic equations of gk D 0 (k D 1; 2; : : : ; mN ) in Eq. (3.8) are gk xk.m/

xk.m/ 1

1 hf ˛1 .xk.m/ C xk.m/ / 1 2

1 ˛2 sinŒ .xk.m/ C xk .m/ / 1 2

1 1 ˇ.xk.m/ C xk.m/ /3 C Q0 cos .t C h/g 1 8 2 D 0:

(3.9)

3.2. FORMULATION FOR PERIOD-m MOTIONS

The time-delay node

xj .m/

xk.m/ D xk.m/ lk

(j D k; k 1

xk.m/ D xk.m/ 1 1 lk

1) in Eq. (3.9) are interpolated by Eq. (3.5)

1 C lk /.xk.m/ xk.m/ /; lk lk 1 h 1 C .1 C lk 1 /.xk.m/ xk.m/ 1 1 lk 1 1 lk h

C .1 1

11

(3.10) 1

/: 1

From Eqs. (3.8)–(3.10), the approximate discrete nodes of period-m motions in the 1-D, time-delayed, nonlinear dynamical system are determined. For the stability and bifurcation of period-m motion, in vicinity of xk.m/ and xk.m/ ; xk.m/ D xk.m/ C xk.m/ and xk.m/ D xk .m/ C xk .m/ . The linearized equations of implicit mappings are k X j Dk 1

with

@gk

@gk

xj.m/ C .m/

@xj

rj D j

. .m/

@xj

lj ; j D k

@xj .m/ @xr.m/ j

xr.m/ C j

@xj .m/ @xr.m/ j 1

xr.m/ / D 0; j 1

(3.11)

1; kI .k D 1; 2; : : : ; mN /:

Deﬁne /T ; D .xk.m/ ; xk.m/1 ; : : : ; xr.m/ y.m/ k k 1

y.m/ D .xk.m/1 ; xk.m/2 ; : : : ; xr.m/ k 1 k 1

T 1/ ;

(3.12)

y.m/ D .xk.m/ ; xk.m/1 ; : : : ; xr.m/ /T ; k k 1

y.m/ D .xk.m/1 ; xk.m/1 ; : : : ; xr.m/ k 1 k 1

1/

T

:

The resultant Jacobian matrix of the period-m motion is

DP D DPmN.mN D

1/;:::;1

D

"

.m/ @ymN

@y0.m/

.m/ .m/ A.m/ mN AmN 1 ; : : : ; A1

# .m/

.y0

.m/

;y1

.m/

;:::;ymN /

(3.13)

D A.m/ ;

where .m/ .m/ .m/ .m/ y.m/ y.m/ D A.m/ 0 mN D A mN AmN 1 ; : : : ; A1 y0 ; " .m/ # @yk .m/ .m/ .m/ .m/ ; yk D Ak yk 1 ; Ak @y.m/ .m/ .m/ k 1 y ;y k 1

k

(3.14)

12

3. PERIODIC MOTIONS IN TIME-DELAY SYSTEMS

and

"

A.m/ D k

.m/ ak.r k

I.m/ k

; s D 1 C lk

1;

s 3

D diag.1; 1; : : : ; 1; 1/ss ; D .0; 0; : : : ; 0; 0/T I „ ƒ‚ … @gk

D

@xk.m/ @gk

.m/ D akr j

ak.rj

1/

1

0.m/ k

s

.m/ akj

#

.sC1/.sC1/ .m/ .m/ .m/ Œak.k 1/ ; 0; : : : ; 0; akrk ; akr ; k 1 „ ƒ‚ …

B.m/ D k I.m/ k 0.m/ k

B.m/ k

!

1

!

@xj 1 j C1 X

@xk.m/

1/

@gk

D

; .m/ @gk @x˛.m/ @x˛ .m/ @xr.m/ j

˛Dj

!

1

j X

@xk.m/

with rj D j

(3.15)

@gk

˛Dj 1

lj ; j D k

;

@gk @x˛.m/ @x˛.m/ @xr.m/ j 1

;

1; kI

and @gk @xk.m/ @gk @xj.m/ @xj .m/ @xr.m/ j 1 D ˛1 M D

1 h; 2

D1

@gk @xk.m/1

@xj .m/

1 D ˛2 h cos M; 2 D

1 h

D

@xr.m/ j

1

D1

1 h; 2 1 C lrj ; h

(3.16)

lrj ;

2 3 .m/ ˇ xk C xk.m/1 ; 4

1 .m/ Œx 2 k lk

1

C xk.m/lk

2

C .1

1 C lk /.xk.m/lk h

xk.m/lk

2

/:

The eigenvalues of DP for the period-m motion in the scalar, time-delayed, nonlinear dynamical system are computed by ˇ ˇ ˇDP I.sC1/.sC1/ ˇ D 0: (3.17) (i) If the magnitudes of all eigenvalues of DP are within the unit circle (i.e., ji j < 1; i D 1; 2; : : : ; .s C 1/), the approximate periodic solution is stable.

3.2. FORMULATION FOR PERIOD-m MOTIONS

13

(ii) If at least the magnitude of one eigenvalue of DP is falling outside of the unit circle (i.e., ji j > 1; i 2 f1; 2; : : : ; .s C 1/g), the approximate periodic solution is unstable. (iii) The boundaries between stable and unstable periodic ﬂow with higher-order singularity give bifurcation and stability conditions with higher-order singularity. The bifurcation conditions are given as follows. (iv) If i D 1 with jj j < 1.i; j 2 f1; 2; : : : ; .s C 1/g and i ¤ j ), the saddle-node bifurcation (SN) occurs. (v) If i D 1 with jj j < 1.i; j 2 f1; 2; : : : ; .s C 1/g and i ¤ j ), the period-doubling bifurcation (PD) occurs. (vi) If ji;j j D 1 with jl j < 1.i; j; l 2 f1; 2; : : : ; .s C 1/g, i D N j , and i ¤ j ¤ l ), the Neimark bifurcation (NB) occurs.

15

CHAPTER

4

A Global Sequential Scenario Periodic motions varying with excitation frequency in the 1-D, time-delayed, dynamical system will be presented from the periodic node displacement xmod.k;N / for mod.k; N / D 0. The periodic nodes are predicted from implicit mappings, and the corresponding stability and bifurcation are determined through the eigenvalue analysis. Consider a set of parameters as ˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0; D T =4;

(4.1)

where T D 2=. The global view of periodic node displacement xmod.k;N / varying with excitation frequency 2 .0; 12:0/ is presented in Fig. 4.1a. The zoomed views for 2 .5:92; 8:40/ and .5:90; 6:38/ are presented in Figs. 4.1b and c, respectively. The solid and dash curves represent stable and unstable periodic motions, respectively. The bold underline letter “S” is for symmetric periodic motions; “A” is for asymmetric periodic motions. The acronyms “P-1” is for period-1 motions,“‘P-2” is for period-2 motions, and so on. “SN” and “PD” are for saddle-node bifurcation and perioddoubling bifurcation, respectively. For saddle-node bifurcations, jumping phenomena or symmetry breaks of period-m motions are observed. For the period-doubling bifurcations, period2m motions appear for period-m motions. In Fig. 4.1, only the base periodic motions are with the following order: P1.S/ G P1.A/ G P3.S/ G P2.A/ G P5.S / G G Pm.A/ G P2mC1.S/ G ;

(4.2)

where Pm.A/ is a fundamental asymmetric period-m (i.e., P -m) motion, and P2mC1.S / is a fundamental symmetric period-.2m C 1/ (i.e., P -.2m C 1/) motion with m D 1; 2; 3; : : :. Based on the fundamental periodic motions, there are two classes of asymmetric period-m motion and symmetric period-(2m C 1) motions. The corresponding bifurcation tree of asymmetric period-m motions to chaos is developed. Such a bifurcation tree to chaos is called the global scenario of the fundamental asymmetric period-m motion. Such a global scenario is expressed by Gm.A/ for the asymmetric period-m motion. The corresponding bifurcation tree of symmetric period-(2m C 1) motion to chaos is also developed. Such a bifurcation tree to chaos is called the global scenario of the fundamental symmetric period-(2m C 1) motion. Such a global scenario is expressed by G2mC1.S/ for the symmetric period-(2m C 1) motions. In such a range of excitation frequency, based on the fundamental symmetric and asymmetric periodic motions, there is a global sequential scenario of periodic motions to chaos as G1.S/ G G1.A/ G G3.S/ G G2.A/ G Gm.A/ G G2mC1.S/ G .m D 1; 2; : : :/;

(4.3)

16

4. A GLOBAL SEQUENTIAL SCENARIO

Figure 4.1: A sequential scenario of bifurcation trees with the base periodic motions. (a) A global view of 2 .0; 12/, (b) a zoomed view of 2 .5:92; 8:40/, and (c) a zoomed view of 2 .5:90; 6:38/. (˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0).

4. A GLOBAL SEQUENTIAL SCENARIO

17

where “S” and “A” are for symmetric and asymmetric periodic motions, respectively. Such global scenarios of the asymmetric and symmetric periodic motions are presented in Figs. 4.2 and 4.3, respectively. The bifurcations of periodic motions in the sequential order of the global bifurcation trees are tabulated in Tables 4.1 and 4.2. Except for the symmetric P-1 motions, all periodic motions in the global sequence have the bifurcation trees to chaos. Herein, focused is the sequences of the bifurcation trees of periodic motions to chaos. In Fig. 4.2, the global scenarios of the asymmetric periodic motions to chaos are presented, and the corresponding bifurcation points are tabulated in Table 4.2. In Fig. 4.2a, the global scenario of asymmetric period-1 motion to chaos is presented for the frequency range of 2 .0; 11:2/, represented by G1.A/ . A zoomed view for period-2 to period-8 motions is placed in a zoomed window. In fact, the stable and unstable asymmetric period-1 motion exist in 2 .10:285; 1/ and .0; 10:285/ for such a set of parameters, respectively. The corresponding period-2 to chaos exists in 2 .0; 10:285/. In other words, the bifurcation scenario of the asymmetric period-1 motion to chaos exists in the frequency range of 2 .0; 1/. In Fig. 4.2b, the detailed view of asymmetric period-1 to period-4 motions are presented. For such an asymmetric period-1 to period-8 motion, there are four (4) period-doubling bifurcations in the bifurcation tree. Continuing period-doubling bifurcations, such a global scenario of the bifurcation tree of the asymmetric period-1 motion will be developed to chaos. In Figs. 4.2c and d, the global scenario of bifurcation trees of the base asymmetric period2 motion to chaos is presented for 2 .0; 7:7577/, represented by G2.A/ . Such a bifurcation tree exists in a ﬁnite range. On both ends of the stable asymmetric period-2 motion, two subbranches of bifurcation trees exist. The right branch is a closed bifurcation tree and the unstable asymmetric period-2 motion. Only stable period-4 motions exist in the right sub-branch, as shown in the zoomed window in Fig. 4.2c. The left and right sub-branches of the bifurcation tree are presented in Fig. 4.2d. The detailed view of the bifurcation trees are clearly presented. For the bifurcation tree of asymmetric period-2 to period-8 motions, one (1) saddle-node bifurcation is for period-2 motion disappearance, and two (2) saddle-node bifurcations are for the period-4 motion jumping phenomena. Three (3) period-double bifurcations are for period-2 to period-4 motion. Three (3) period-doubling bifurcations are also for period-4 to period-8 motions, and four (4) period-doubling bifurcations are for period-8 to period-16 motions. In Figs. 4.2e and f, the global scenario of bifurcation trees of the base asymmetric period3 motion to chaos is presented for 2 .0; 6:6003/, represented by G3.A/ . Such a bifurcation tree also exists in a ﬁnite range. On both ends of the stable asymmetric period-3 motion, two sub-branches of bifurcation trees exist. The right branch of such a bifurcation tree is zoomed, as shown in the zoomed window in Fig. 4.2e. The left and right sub-branches of the bifurcation tree are zoomed, as shown in Fig. 4.2f. The detailed view of the bifurcation tree is clearly observed. For the bifurcation tree of asymmetric period-3 to period-6 motions, one (1) saddle-node bifurcation is for period-3 motion disappearance, and four (4) saddle-node bifurcations are for the

18

4. A GLOBAL SEQUENTIAL SCENARIO

Figure 4.2: Bifurcation trees based on asymmetric period-m motion. G1.A/ : (a) 2 .0; 11:2/ and (b) 2 .9:8; 10:4/; G2.A/ : (c) 2 .0; 8:0/ and (d) 2 .6:90; 7:10/; G3.A/ : (e) 2 .0; 6:8/; and (f ) 2 .6:30; 6:62/; G4.A/ : (g) 2 .0; 6:4/ and (h) 2 .6:06; 6:24/; G5.A/ : (i) 2 .0; 6:4/; and (j) 2 .5:935; 6:040/. (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). (Continues.)

4. A GLOBAL SEQUENTIAL SCENARIO

19

Figure 4.2: (Continued.) Bifurcation trees based on asymmetric period-m motion. G1.A/ : (a) 2 .0; 11:2/ and (b) 2 .9:8; 10:4/; G2.A/ : (c) 2 .0; 8:0/; and (d) 2 .6:90; 7:10/; G3.A/ : (e) 2 .0; 6:8/; and (f ) 2 .6:30; 6:62/; G4.A/ : (g) 2 .0; 6:4/ and (h) 2 .6:06; 6:24/; G5.A/ : (i) 2 .0; 6:4/; and (j) 2 .5:935; 6:040/. (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

20

4. A GLOBAL SEQUENTIAL SCENARIO

Figure 4.3: Bifurcation trees based on symmetric period-m motion. G3.S/ : (a) 2 .0; 10:0/ and (b) 2 .7:6; 10:0/; G5.S/ : (c) 2 .0:0; 7:6/ and (d) 2 .6:55; 6:64/; G7.S / : (e) 2 .0; 6:8/ and (f ) 2 .6:16; 6:40/; G9.S / : (g) 2 .0; 6:4/ and (h) 2 .6:00; 6:12/; G11.S/ : (i) 2 .0; 6:4/; and () 2 .5:90; 5:98/. (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). (Continues.)

4. A GLOBAL SEQUENTIAL SCENARIO

21

Figure 4.3: (Continued.) Bifurcation trees based on symmetric period-m motion. G3.S / : (a) 2 .0; 10:0/ and (b) 2 .7:6; 10:0/; G5.S/ : (c) 2 .0:0; 7:6/ and (d) 2 .6:55; 6:64/; G7.S/ : (e) 2 .0; 6:8/ and (f ) 2 .6:16; 6:40/; G9.S/ : (g) 2 .0; 6:4/ and (h) 2 .6:00; 6:12/; G11.S/ : (i) 2 .0; 6:4/; and (j) 2 .5:90; 5:98/. (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

22

4. A GLOBAL SEQUENTIAL SCENARIO

Table 4.1: The bifurcations of asymmetric periodic motions to chaos .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/. (Continues.) 10.2850 9.9520 9.9479 9.9470

Bifurcations PD PD PD PD

G2(A) (0, 7.7577)

7.7580000 7.7480000 7.7048000 7.0573000 6.9755000 6.9749000 6.9738088 7.0284810 7.0284786 7.0284777 6.9710224 6.9710223 7.0017829

SN PD PD PD PD PD PD PD SN SN PD PD PD

G3(A) (0, 6.6003)

6.6002850 6.6001800 6.6001250 6.5570000 6.5569904 6.5584623 6.5576000 6.5550000 6.5490000 6.3510000 6.3727000

SN PD PD PD SN SN PD PD PD PD PD

Frequency Range G1(A) (0, ∞)

Ωcr

Motion Switching P-1 (A) to P-2 (A) P-2 (A) to P-4 (A) P-4 (A) to P-8 (A) P-8 (A) to P-16 (A) G1(A) - Chaos (A) (2 sets or more) P-2 (A) vanishing/appearance P-2 (A) to P-4 (A) P-2 (A) to P-4 (A) P-2 (A) to P-4 (A) P-4 (A) to P-8 (A) P-8 (A) to P-16 (A) P-8 (A) to P-16 (A) P-4 (A) to P-8 (A) P-4 (A) jumping P-4 (A) jumping P-4 (A) to P-8 (A) P-8 (A) to P-16 (A) P-8 (A) to P-16 (A) G2(A) - Chaos (A) (2 sets or more) P-3 (A) vanishing/appearance P-3 (A) to P-6 (A) P-6 (A) to P-12 (A) P-6 (A) to P-12 (A) P-6 (A) jumping P-6 (A) jumping P-6 (A) to P-12 (A) P-6 (A) to P-12 (A) P-3 (A) to P-6 (A) P-3 (A) to P-6 (A) P-6 (A) to P-12 (A)

4. A GLOBAL SEQUENTIAL SCENARIO

Table 4.1: (Continued.) The bifurcations of asymmetric periodic motions to chaos .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/. (Continues.) 6.3433400 6.3432577 6.3800155 6.3800154

PD SN SN PD

G4(A) (0, 6.2124)

6.2125000 6.2120000 6.2123400 6.1847055 6.1847051 6.1884059 6.1883000 6.1830000 6.1803000 6.1140000 6.1085000 6.0891600 6.0891460 6.1127692 6.1127691

SN PD PD PD SN SN PD PD PD PD PD PD SN SN PD

G5(A) (0, 6.027498)

6.0274980 6.0274940 6.0274941 6.0093723 6.0093722 6.0128199 6.0128140 6.0081000 6.0060000 5.9730000

SN PD PD PD SN SN PD PD PD PD

P-6 (A) to P-12 (A) P-6 (A) jumping P-6 (A) jumping P-6 (A) to P-12 (A) G3(A) - Chaos (A) (2 sets or more) P-4 (A) jumping P-4 (A) to P-8 (A) P-8 (A) to P-16 (A) P-8 (A) to P-16 (A) P-8 (A) jumping P-8 (A) jumping P-8 (A) to P-16 (A) P-8 (A) to P-16 (A) P-4 (A) to P-8 (A) P-4 (A) to P-8 (A) P-8 (A) to P-16 (A) P-8 (A) to P-16 (A) P-8 (A) jumping P-8 (A) jumping P-8 (A) to P-16 (A) G4(A) - Chaos (A) (2 sets or more) P-5 (A) vanishing/appearance P-5 (A) to P-10 (A) P-10 (A) to P-20 (A) P-10 (A) to P-20 (A) P-10 (A) jumping P-10 (A) jumping P-10 (A) to P-20 (A) P-10 (A) to P-20 (A) P-5 (A) to P-10 (A) P-5 (A) to P-10 (A)

23

24

4. A GLOBAL SEQUENTIAL SCENARIO

Table 4.1: (Continued.) The bifurcations of asymmetric periodic motions to chaos .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/. 5.9769900 5.9572390 5.9572320 5.9729760 5.9729750

PD PD SN SN PD

PD- for Period-doubling, SN- for saddle-node.

P-10 (A) to P-20 (A) P-10 (A) to P-20 (A) P-10 (A) jumping P-10 (A) jumping P-10 (A) to P-20 (A) G5(A) - Chaos (A) (2 sets or more)

4. A GLOBAL SEQUENTIAL SCENARIO

Table 4.2: The bifurcations of symmetric periodic motions to chaos .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/. (Continues). Frequency Range G1(S) (0, 5.6538) G3(S) (0, 9.7527)

G5(S) (0, 6.992911)

5.6538 9.7527 7.912 7.7764 7.827574 7.827546 7.82753 7.767076 7.767075 7.79126 7.791259

Bifurcations SN SN SN SN SN PD PD PD SN SN PD

6.9929110 6.9924500 6.9919000 6.9918000 6.9411000 6.9405000 6.9293000 6.6333000 6.6120000 6.6070000 6.5982000 6.5981380 6.6025553 6.6025500 6.5761400 6.5758760 6.6216774 6.6216769 6.6216763 6.5746895

SN SN PD PD PD PD SN SN PD PD PD SN SN PD PD SN SN PD PD PD

Ωcr

Motion Switching P-1 (S) vanishing/appearance P-3 (S) vanishing/appearance P-3 (S) symmetry break P-3 (A) jumping P-3 (A) jumping P-3 (A) to P-6 (A) P-6 (A) to P-12 (A) P-6 (A) to P-12 (A) P-6 (A) jumping P-6 (A) jumping P-6 (A) to P-12 (A) G3(S) - Chaos (2 sets or more) P-5 (S) vanishing/appearance P-5 (S) symmetry break P-5 (A) to P-10 (A) P-10 (A) to P-20 (A) P-10 (A) to P-20 (A) P-5 (A) to P-10 (A) P-5 (S) symmetry break P-5 (S) symmetry break P-5 (A) to P-10 (A) P-10 (A) to P-20 (A) P-10 (A) to P-20 (A) P-10 (A) jumping P-10 (A) jumping P-10 (A) to P-20 (A) P-5 (A) to P-10 (A) P-5 (A) jumping P-5 (A) jumping P-5 (A) to P-10 (A) P-10 (A) to P-20 (A) P-10 to P-20

25

26

4. A GLOBAL SEQUENTIAL SCENARIO

Table 4.2: (Continued.) The bifurcations of symmetric periodic motions to chaos .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/. (Continues). 6.5746896 6.6020713 6.6020712

SN SN PD

G7(S) (0, 6.365874)

6.365874 6.36584 6.36581 6.331213 6.331212 6.334354 6.3341 6.3288 6.3257 6.2252 6.2165 6.1933 6.192978 6.2223499 6.22235

SN SN PD PD SN SN PD PD SN SN PD PD SN SN PD

G9(S) (0, 6.1054676)

6.1054676 6.105461 6.1054543 6.08317 6.08316 6.0868624 6.08684 6.0831695 6.0801 6.0332 6.029 6.013866 6.0138554

SN SN PD PD SN SN PD PD SN SN PD PD SN

P-10 (A) jumping P-10 (A) jumping P-10 (A) to P-20 (A) G7(S) - Chaos (4 sets or more) P-7 (S) vanishing/appearance P-7 (A) symmetry break P-7 (A) to P-14 (A) P-7 (A) to P-14 (A) P-7 (A) jumping P-7 (A) jumping P-7 (A) to P-14 (A) P-7 (A) to P-14 (A) P-7 (S) symmetry break P-7 (S) symmetry break P-7 (A) to P-14 (A) P-7 (A) to P-14 (A) P-7 (A) jumping P-7 (A) jumping P-7 (A) to P-14 (A) G7(S) - Chaos (4 sets or more) P-9 (S) vanishing/appearance P-9 (S) symmetry break P-9 (A) to P-18 (A) P-9 (A) to P-18 (A) P-9 (A) jumping P-9 (A) jumping P-9 (A) to P-18 (A) P-9 (A) to P-18 (A) P-9 (S) symmetry break P-9 (S) symmetry break P-9 (A) to P-18 (A) P-9 (A) to P-18 (A) P-9 (A) jumping

4. A GLOBAL SEQUENTIAL SCENARIO

Table 4.2: (Continued.) The bifurcations of asymmetric periodic motions to chaos .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/.

G11(S) (0, 5.968659)

6.03304044 6.03304042

SN PD

5.968659 5.96866 5.96866 5.953775 5.95379 5.9568887 5.95688 5.9528 5.952 5.926 5.9237 5.91342 5.913415 5.9264673 5.926467

SN SN PD PD SN SN PD PD SN SN PD PD SN SN PD

PD-for period-doubling, SN-for saddle-node.

P-9 (A) jumping P-9 (A) to P-18 (A) G9(S) - Chaos (4 sets or not) P-11 (S) vanishing/ appearance P-11 (S) symmetry break P-11 (A) to P-22 (A) P-11 (A) to P-22 (A) P-11 (A) jumping P-11 (A) jumping P-11 (A) to P-22 (A) P-11 (A) to P-22 (A) P-11 (S) symmetry break P-11 (S) symmetry break P-11 (A) to P-22 (A) P-11 (A) to P-22 (A) P-11 (A) jumping P-11 (A) jumping P-11 (A) to P-22 (A) G11(S) - Chaos (4 sets or more)

27

28

4. A GLOBAL SEQUENTIAL SCENARIO

period-6 motion jumping phenomena. Two (2) period-double bifurcations are for period-3 to period-6 motion. Seven (7) period-doubling bifurcations are for period-6 to period-12 motions. In Figs. 4.2g and h, the global scenario of bifurcation trees of the base asymmetric period4 motion to chaos is presented for 2 .0; 6:2124/, represented by G4.A/ . On both ends of the stable asymmetric period-4 motion, two sub-branches of bifurcation trees exist. Such asymmetric period-4 and period-8 motions are diﬀerent from the period-4 and period-8 motions in the bifurcation tress of both asymmetric period-1 and period-2 motions. The right branch of such a bifurcation tree is in the zoomed window in Fig. 4.2g. The left and right sub-branches of the bifurcation tree are zoomed, as shown in Fig. 4.2h. The detailed view of the bifurcation tree is clearly observed. For the bifurcation tree of asymmetric period-4 to period-8 motions. One (1) saddle-node bifurcation is for period-4 motion disappearance, and four (4) saddle-node bifurcations are for the period-8 motion jumping phenomena. Three (3) period-double bifurcations are for period-4 to period-8 motion. Seven (7) period-doubling bifurcations are for period-4 to period-8 motions. In Figs. 4.2i and j, the global scenario of bifurcation trees of the base asymmetric period5 motion to chaos is presented for 2 .0; 6:027498/, represented by G5.A/ . On both ends of the stable asymmetric period-5 motion, two sub-branches of the bifurcation tree exist. The right branch of such a bifurcation tree is zoomed, as shown in Fig. 4.2i. The left and right sub-branches of the bifurcation tree are zoomed, as shown in Fig. 4.2i. The detailed view of the bifurcation tree is clearly observed. For the bifurcation tree of asymmetric period-5 to period-10 motions, One (1) saddle-node bifurcation is for period-5 motion vanishing, and four (4) saddle-node bifurcations are for the period-10 motion jumping phenomena. Three (3) period-double bifurcations are for period-5 to period-10 motion. Seven (7) period-doubling bifurcations are for period-10 to period-20 motions. Based on the bifurcation numbers and properties, the bifurcation trees of the fundamental asymmetric period-3, period-4, and period-5 motions have the similar bifurcation structures. In Fig. 4.1, the order of the fundamental periodic motions was presented. The base symmetric periodic motions are embedded with the base asymmetric periodic motions. For the symmetric period-1 motion, no bifurcation tree is observed for such speciﬁc parameters. The symmetric period-1 motion exist for 2 .0; 5:6538/ with a saddle-node bifurcation of cr 5:6538. Such a saddle-node bifurcation is for the symmetric period-1 motion vanishing. Thus, the global scenario of the base symmetric period-3 motions to chaos is presented for 2 .0; 9:7527/ in Figs. 4.3a and b ﬁrst. Such a global scenario is represented by G3.S / G21C1.S/ , which is generated from the asymmetric period-1 motion, because a symmetric period-1 motion has two sets of asymmetric period-1 motion solutions. With adding one solution branch, the two sets of asymmetric period-1 motion can easily jump to the symmetric period-3 motions with three solution branches. At the right end of the symmetric period-3 motion, no any sub-branch of bifurcation trees exists. Only there is a saddle-node bifurcation for symmetric period-3 motion vanishing. The unstable symmetric period-3 motion goes back to

4. A GLOBAL SEQUENTIAL SCENARIO

29

D 0 from cr 5:6538. At the left end of the stable symmetric period-3 motions, there is a saddle-node bifurcation, and the symmetric period-3 motion becomes two asymmetric period-3 motions. Through the two asymmetric period-3 motions, there are two branches of the bifurcation trees of asymmetric period-3 motions to chaos. The zoomed view of such bifurcation tree is shown in Fig. 4.3a. In Fig. 4.3b, the local view of the symmetric period-3 motion to asymmetric period-3 motions is presented. The bifurcation structure in the bifurcation tree is clearly observed. For the bifurcations tree of the base symmetric period-3 motion, one (1) saddle-node bifurcation is for symmetric period-3 motion vanishing. One (1) saddle-node bifurcation is for a symmetry-break of the symmetric to asymmetric period-3 motion. Two (2) saddle-node bifurcations are for asymmetric period-3 motion jumping. Two (2) saddle-node bifurcations are for asymmetric period-6 motion jumping. One (1) period-doubling bifurcation is for asymmetric period-3 to period-6 motion. Three (3) period-doubling bifurcations are for period-6 to period12 motions. Continuing such period-doubling bifurcations, the bifurcation tree of symmetric period-3 motions to chaos. Such a global scenario of the symmetric period-3 motion to chaos is diﬀerent from the asymmetric period-3 motion to chaos. After symmetric period-3 motions, from the sequential order of periodic motions, asymmetric period-2 motions can be observed, which was presented as before. With four solution branches of the asymmetric period-2 motions, the symmetric period-5 motion can be formed if adding one solution branch. The global scenario of the symmetric period-5 motion to chaos (G5.S/ ) is presented for 2 .0; 6:992911/ in Figs. 4.3c and d. The global view of such a global scenario is placed in Fig. 4.3c but the bifurcation tree is not clear. The zoomed view of the right end of the symmetric period-5 motion is also placed in Fig. 4.3c. At the right end, the sub-branch of the bifurcation tree is closed. At the left end of the symmetric period-5 motion, the sub-branch of bifurcation tree is open. One (1) saddle-node bifurcation is for the symmetric period-5 motion vanishing. Three (3) saddle-node bifurcations are for the symmetry break of the symmetric period-5 motion. Two (2) saddle-node bifurcations are for the asymmetric period-5 motion jumping. Four (4) saddle-node bifurcations are for the asymmetric period-10 motion jumping. Five (5) period-doubling bifurcations are for asymmetric period-5 to period-10 motions. Eight (8) period-doubling bifurcations are for the period-10 to period-20 motions. As for the symmetric period-5 motion, the global scenario of the symmetric period-7 motion to chaos (G7.S/ ) is presented for 2 .0; 6:365874/ in Figs. 4.3e and f. The global view of such a global scenario is placed in Fig. 4.3d, and the zoomed view of the right end of the symmetric period-7 motion is also arranged. At the right end, the sub-branch of the bifurcation tree of the symmetric period-7 motion is closed. At the left end of the symmetric period-7 motion, the sub-branch of the bifurcation tree is open, and the corresponding unstable periodic motions reach to D 0. One (1) saddle-node bifurcation is for the symmetric period-7 motion vanishing. Three (3) saddle-node bifurcations are for the symmetry break of the symmetric period-7 motion. Four (4) saddle-node bifurcations are for the asymmetric period-7 motion jumping. Seven (7) period-doubling bifurcations are for asymmetric period-7 to period-14 motions.

30

4. A GLOBAL SEQUENTIAL SCENARIO

Similarly, in Figs. 4.3g and h, presented is a bifurcation tree of the symmetric period-9 motion to chaos (G9.S/ ) for 2 .0; 6:1054676/. In Fig. 4.3g, the global view of such a bifurcation tree is presented, and the right end of the symmetric period-9 motion is zoomed to show the detailed bifurcations. The detailed bifurcation tree near the symmetric period-9 motion is zoomed, as shown in Fig. 4.3h. At the right end, the sub-branch of the bifurcation tree of the symmetric period-9 motion is closed. The symmetric period-9 motion goes to the bifurcation tree and then return back to the symmetric period-9 motion. At the left end of the symmetric period-9 motion, the sub-branch of the bifurcation tree is open, and the corresponding unstable periodic motions also reach to D 0. One (1) saddle-node bifurcation is for the symmetric period-9 motion vanishing. Three (3) saddle-node bifurcations are for the symmetry break of the symmetric period-9 motion. Four (4) saddle-node bifurcations are for the asymmetric period-9 motion jumping. Seven (7) period-doubling bifurcations are for asymmetric period-9 to period18 motions. For the global scenario of G11.S/ , a bifurcation tree of the symmetric period-11 motion to chaos for 2 .0; 5:968659/ is similar to the symmetric period-9 motion, as shown in Figs. 4.3i and j. The zoomed, closed bifurcation tree at the right end of the symmetric period-11 motion is illustrated in Fig. 4.3i. The detailed bifurcation tree near the symmetric period-11 motion is shown in Fig. 4.3j. At the left end of the symmetric period-11 motion, the sub-branch of the bifurcation tree is also open. The bifurcations in the global scenario of the symmetric period-11 motion are similar to the bifurcation tree of the symmetric period-9 motion. One (1) saddlenode bifurcation is for the symmetric period-11 motion vanishing. Three (3) saddle-node bifurcations are for the symmetry break of the symmetric period-11 motion. Four (4) saddle-node bifurcations are for the asymmetric period-11 motion jumping. Seven (7) period-doubling bifurcations are for asymmetric period-11 to period-22 motions.

31

CHAPTER

5

Frequency-Amplitude Analysis In this chapter, from the nodes of period-m motions, the corresponding nonlinear frequencyamplitude characteristics of the global sequential period-m motions to chaos are determined from the ﬁnite Fourier series.

5.1

FINITE FOURIER SERIES

Consider the nodes of period-m motions as xk.m/ for k D 0; 1; 2; : : : ; mN in the 1-D, timedelayed, nonlinear system. The approximate expression of a period-m motion is determined by the ﬁnite Fourier series as x .m/ .t/ a0.m/ C

M X

bj=m cos.

j D1

j j t / C cj=m sin. t/: m m

(5.1)

There are .2M C 1/ unknown vector coeﬃcients of a0.m/ ; bj=m ; cj=m .j D 1; 2; : : : ; M /. From the given nodes xk.m/ .k D 0; 1; 2; : : : ; mN /, such unknowns (2M C 1 mN C 1) can be determined. In other words, M mN=2. The node points xk.m/ on the period-m motion can be expressed by the ﬁnite Fourier series as for tk 2 Œ0; mT x

.m/

.tk /

x .m/ k

D

D a0.m/ C

a0.m/ mN=2 X

C

mN=2 X

bj=m cos.

j D1

bj=m cos.

j D1

.k D 0; 1; : : : ; mN

j j tk / C cj=m sin. tk / m m

j 2k j 2k / C cj=m sin. / m N m N

(5.2)

1/;

where T D

2 D Nt I

tk D kt D

2k : N

(5.3)

32

5. FREQUENCY-AMPLITUDE ANALYSIS

From discrete nodes on the period-m motion, Equation (5.1) gives a0.m/ D

mN 1 1 X .m/ xk ; N kD0

bj=m D

2 mN

cj=m D

2 mN

mN P1 kD1 mN P1 kD1

xk.m/

2j cos.k mN /;

xk.m/

2j / sin.k mN

9 > > > = > > > ;

(5.4) .j D 1; 2; : : : ; mN=2/:

Thus, the approximate expression of a period-m motion in Eq. (3.1) is determined by x .m/ .t/ a0.m/ C

mN=2 X

bj=m cos.

j D1

j j t / C cj=m sin. t/: m m

(5.5)

The foregoing equation can be rewritten as x .m/ .t/ a0.m/ C

mN=2 X j D1

Aj=m1 cos.

j t m

'j=m /;

where the harmonic amplitudes and harmonic phases of the period-m motion are q cj=m 2 2 : Aj=m D bj=m C cj=m ; 'j=m D arctan bj=m

5.2

(5.6)

(5.7)

FREQUENCY-AMPLITUDE CHARACTERISTICS

From the semi-analytical prediction, the discrete nodes of periodic motions are obtained from the corresponding mapping structures. Such discrete nodes can be used for discrete Fourier analysis, and the frequency-amplitude characteristics of periodic motions can be analyzed. The solid and dash curves represent stable and unstable periodic motions, respectively. The bold underline letter “S” is for symmetric periodic motions; “A” is for asymmetric periodic motions. The acronyms “P-m” is for period-m motions. “SN” and “PD” is for saddle-node bifurcation and period-doubling bifurcation, respectively. For the symmetric period-1 motion, the solution is very simple. Thus, the frequencyamplitude analysis of such a symmetric period-1 motion will not be done herein. Based on the sequential order of the global scenarios of bifurcation trees of periodic motions to chaos, the constant terms for the bifurcation trees relative to the asymmetric period-1 motion to the symmetric period-11 motion are presented in Figs. 5.1a–j. For the symmetric periodic motions, the constant terms a0.m/ D 0 but for the asymmetric periodic motion, a0.m/ ¤ 0. The global views of bifurcation trees of asymmetric period-1 motions to the symmetric period-11 motion are presented. The bifurcation developing intervals are very crowed.

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

33

Figure 5.1: Constant terms for bifurcation trees. (a) G1.A/ , (b) G3.S/ , (c) G2.A/ , (d) G5.S/ , (e) G3.A/ , (f ) G7.S / , (g) G4.A/ , (h) G9.S / , (i) G2.A/ , (j) G11.S / . (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). (Continues.)

34

5. FREQUENCY-AMPLITUDE ANALYSIS

Figure 5.1: (Continued.) Constant terms for bifurcation trees. (a) G1.A/ , (b) G3.S / , (c) G2.A/ , (d) G5.S/ , (e) G3.A/ , (f ) G7.S/ , (g) G4.A/ , (h) G9.S/ , (i) G2.A/ , (j) G11.S/ . (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

5.2.1 ASYMMETRIC PERIOD-1 TO PERIOD-8 MOTIONS For the further studies of nonlinear frequency-amplitude characteristics, the frequencyamplitude characteristics of the global scenario of the bifurcation trees of the asymmetric period1 motion to chaos are presented in Fig. 5.2. In Fig. 5.2a, a global view of constant terms for the asymmetric period-1 motion to chaos is presented through period-1 to period-8 motion. The zoomed area of the bifurcation tree is presented in Fig. 5.2b. The asymmetric period-2 to period-8 motions are obviously illustrated. In Fig. 5.2c, the harmonic amplitude of A1=8 is for the period-8 motion. In Fig. 5.2d, the harmonic amplitudes of A1=4 are for the period-4 and period-8 motions. In Fig. 5.2e, the harmonic amplitude of A3=8 is for the period-8 motion, which similar to A1=8 . In Fig. 5.2f, the harmonic amplitudes of A1=2 are for the period-2, period-4 and period-8 motions. In Figs. 5.2g and h, are presented the global and zoomed views of the harmonic amplitude of A1 for period-1 to period-8 motions, respectively. The bifurcation

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

35

details between two periodic motions are observed. Similarly, in Figs. 5.2i and j, presented are the global and zoomed views of the harmonic amplitude of A2 for period-1 to period-8 motions, respectively. The bifurcation tree presented through A1 and A2 are diﬀerent. To avoid abundant illustrations, the zoomed views of harmonic amplitudes A19 and A20 for asymmetric period-1 to period-8 motions are presented in Figs. 5.2k and l. The quantity levels of the two harmonic amplitudes are down to 10 12 for frequency near D 10:2850.

5.2.2 ASYMMETRIC PERIOD-2 TO PERIOD-8 MOTIONS The frequency-amplitude characteristics of the bifurcation tree of the asymmetric period-2 motion to chaos are presented in Fig. 5.3. The bifurcation tree of the asymmetric period-2 motion exists in the ﬁnite frequency range of 2 .0; 7:7577/. A global view of constant terms for the asymmetric period-2 motion to chaos is presented in Fig. 5.3a. The zoomed view of the bifurcation tree is presented in Fig. 5.3b, and the period-2 to period-8 motions are obviously illustrated. At the right of the stable asymmetric period-2 motion, the stable period-4 motion in the right bifurcation tree is observed, which is connected with stable asymmetric period-2 motion. Such a stable period-2 motion with saddle-node bifurcation become unstable, and the unstable asymmetric period-2 motion returns back to D 0. In Fig. 5.3c, the harmonic amplitude of A1=8 is for the period-8 motion with two branches. In Fig. 5.3d, the harmonic amplitudes of A1=4 are for period-4 and period-8 motions in the bifurcation tree, which are diﬀerent from the bifurcation tree relative to the asymmetric period-1 motion. In Fig. 5.3e, the harmonic amplitude of A3=8 is also for the period-8 motion with two branches. In Fig. 5.3f, the harmonic amplitudes of A1=2 are for the period-2, period-4, and period-8 motions. In Figs. 5.3g and h, the global and zoomed views of the harmonic amplitude of A1 are for period-2 to period-8 motions, respectively. The bifurcation details between two periodic motions are observed. Similarly, in Figs. 5.3i and j, presented are the global and zoomed views of the harmonic amplitude of A2 for period-2 to period-8 motions, respectively. To avoid abundant illustrations, the zoomed views of harmonic amplitudes A19 and A20 for asymmetric period-2 to period-8 motions are presented in Figs. 5.3k and l. The quantity levels of the two harmonic amplitudes are down to 10 7 for frequency near D 7:7577. 5.2.3 ASYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS The bifurcation tree of the asymmetric period-3 motion to chaos is presented through the period3 and period-6 motions in Fig. 5.4. Such a bifurcation tree of the asymmetric period-3 motion is in the ﬁnite frequency range of 2 .0; 6:6003/. In Figs. 5.4a and b, the global and zoomed views of the constant terms for the bifurcation tree of the asymmetric period-3 motions, respectively, are presented. At the right end of the asymmetric period-3 motion, the asymmetric period-6 motion has two period-doubling bifurcations to return back to the stable asymmetric period3 motion. The asymmetric stable period-3 motion has a saddle-node bifurcation to become unstable and returns back to D 0. At the left end of the asymmetric period-3 motion, the

36

5. FREQUENCY-AMPLITUDE ANALYSIS

Figure 5.2: Frequency-amplitude characteristics for a global bifurcation tree based on asymmetric-period-1 motions G1.A/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 1; 2; 3; 4; 8; 16; 152; 160; m D 8) (˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0). (Continues.)

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

37

Figure 5.2: (Continued.) Frequency-amplitude characteristics for a global bifurcation tree based on asymmetric-period-1 motions G1.A/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 1; 2; 3; 4; 8; 16; 152; 160; m D 8) (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0; Q0 D 3:0).

38

5. FREQUENCY-AMPLITUDE ANALYSIS

Figure 5.3: Frequency-amplitude characteristics for a global bifurcation tree based on asymmetric-period-2 motions G2.A/ . (a) and b constant a0.m/ , c–l Ak=m (k D 1; 2; 3; 4; 8; 16; 152; 160; m D 8) (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). (Continues.)

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

39

Figure 5.3: (Continued.) Frequency-amplitude characteristics for a global bifurcation tree based on asymmetric-period-2 motions G2.A/ . (a) and b constant a0.m/ , c–l Ak=m (k D 1; 2; 3; 4; 8; 16; 152; 160; m D 8) (˛1 D 10:0; ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

40

5. FREQUENCY-AMPLITUDE ANALYSIS

bifurcation tree of the asymmetric period-3 motion are open to D 0. In Figs. 5.4c and e, the harmonic amplitude A1=6 and A1=2 is for the period-6 motion, and the bifurcations of period-6 motions are very clearly illustrated. In Figs. 5.4d and f, the harmonic amplitudes A1=3 and A2=3 are for the asymmetric period-3 and period-6 motions. In Figs. 5.4g and h, the global and local views of the bifurcation tree based on the harmonic amplitude of A1 are presented. The quantity levels of Ak=6 (k D 1; 2; 3; 4; 6) are over 0.1. The harmonic amplitudes of A2 for period-3 and period-6 motions are presented in Figs. 5.4i and j, and the corresponding quantity level is below 0.1. In Figs. 5.4k and l, the harmonic amplitudes of A25 and A26 drop down to 10 8 for the frequency near D 6:6003. For the bifurcation trees of asymmetric period-4 and period-5 motions, the frequencyamplitude characteristics are similar to the asymmetric period-3 motion, and the bifurcation points are similar too, as tabulated in Table 4.1. In the sequential order of the global scenarios of the bifurcation trees, the asymmetric periodic motions are embedded with the symmetric periodic motions. Because the symmetric period-1 motion possesses simple scenarios without the bifurcations for the prescribed parameters. Thus, the frequency amplitude analysis of the bifurcation trees of the symmetric period-3 motion is discussed ﬁrst.

5.2.4 SYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS In Fig. 5.5, the scenario of the bifurcation tree of the symmetric period-3 motion is presented, which is diﬀerent from the asymmetric period-3 motion. In Figs. 5.5a and b, the global and local views of the constant terms for such a global scenario of the bifurcation tree are presented. For the symmetric period-3 motion, the constant term is a0.3/ D 0 but for asymmetric period-5 and period-10 motions, the constant terms are a0.3/ ¤ 0 and a0.6/ ¤ 0. The two asymmetric period3 and period-6 motions are observed. At the right end of the symmetric period-3 motion, no bifurcation tree is formed. At the left end of the symmetric period-3 motion, the bifurcation tree is open and goes to the frequency of D 0. The stable symmetric period-6 motion exist in the very short frequency range. In Figs. 5.5c and d, the global and local views of the harmonic amplitudes of A1=3 are presented for the period-3 and period-6 motions. In Figs. 5.5e and f, the global and local views of the harmonic amplitudes of A2=3 are presented for the period-3 and period-6 motions. For the symmetric period-3 motion, the harmonic amplitude is A2=3 D 0. In Figs. 5.5g and h, the global and local views of the harmonic amplitudes of A1 are presented for the period-3 and period-6 motions, which are similar to the harmonic amplitudes of A1=3 with the same quantity level. In Figs. 5.5i and j, the global and local views of the harmonic amplitudes of A2 are also presented for the period-3 and period-6 motions, which are similar to the harmonic amplitudes of A2=3 with the smaller quantity level. In Figs. 5.5k and l, the harmonic amplitudes of A18 and A19 drop down to 10 8 for the frequency near D 9:7527.

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

41

Figure 5.4: Frequency-amplitude characteristics for a global bifurcation tree G3.A/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 1; 2; 3; 4; 6; 12; 150; 156; m D 6) (˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0). (Continues.)

42

5. FREQUENCY-AMPLITUDE ANALYSIS

Figure 5.4: (Continued.) Frequency-amplitude characteristics for a global bifurcation tree G3.A/ . (a) and b constant a0.m/ , c–l Ak=m (k D 1; 2; 3; 4; 6; 12; 150; 156; m D 6) (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

43

Figure 5.5: Frequency-amplitude characteristics for a global bifurcation tree G3.S/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 2; 4; 6; 12; 108; 114; m D 6) (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). (Continues.)

44

5. FREQUENCY-AMPLITUDE ANALYSIS

Figure 5.5: (Continued.) Frequency-amplitude characteristics for a global bifurcation tree G3.S/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 2; 4; 6; 12; 108; 114; m D 6) (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

45

5.2.5 SYMMETRIC PERIOD-5 TO PERIOD-10 MOTIONS The global scenario of the symmetric period-5 motion to chaos is presented for 2 .0; 6:992911/ in Fig. 5.6. In Figs. 5.6a and b, the global and zoomed views of the constant terms for the bifurcation tree of the symmetric period-5 motions, respectively, are presented. For the symmetric period-5 motion, the constant term is a0.5/ D 0 but for asymmetric period-5 and period-10 motions, the constant terms are a0.5/ ¤ 0 and a0.10/ ¤ 0. The symmetric structure of the constant terms is observed. At the right end of the symmetric period-5 motion, the symmetric period-10 motion has two period-doubling bifurcations to return back to the stable symmetric period-10 motion. The symmetric stable period-10 motion has a saddle-node bifurcation to become unstable and return back to D 0. At the left end of the symmetric period-5 motion, the bifurcation tree of the symmetric period-5 motion are open to D 0. In Figs. 5.6c and e, the harmonic amplitudes A1=10 and A1=5 are for the period-5 and period-10 motions, and the bifurcation trees are clearly illustrated. In Figs. 5.6d and f, the harmonic amplitudes A3=10 and A2=5 are for the asymmetric period-5 and period-10 motions. In Figs. 5.6g and h, the global and local views of the bifurcation tree of the symmetric period-5 motion are presented through the harmonic amplitudes of A1 . The harmonic amplitudes of A2 for period-5 and period-10 motions are presented in Figs. 5.6i and j. In Figs. 5.6k and l, the harmonic amplitudes of A29 and A30 drop down to 10 10 for the frequency near D 6:992911. For the bifurcation trees of symmetric period-7, period-9, and period-11 motions, the frequency-amplitude characteristics are similar to the bifurcation tree of the asymmetric period5 motion, and the corresponding bifurcation points are similar, as listed in Table 4.2. From the frequency-amplitude analysis, the bifurcation tree is much simpler than the periodic nodes points vs. excitation frequency. In addition, the quantity levels of the harmonic amplitudes are determined, and the accuracy of the periodic motions can be determined.

46

5. FREQUENCY-AMPLITUDE ANALYSIS

Figure 5.6: Frequency-amplitude characteristics for a global bifurcation tre G5.S/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 2; 4; : : : ; 10; 20; 190; 200; m D 10) (˛1 D 10:0; ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). (Continues.)

5.2. FREQUENCY-AMPLITUDE CHARACTERISTICS

47

Figure 5.6: (Continued.) Frequency-amplitude characteristics for a global bifurcation tre G5.S/ . (a) and (b) constant a0.m/ , (c)–(l) Ak=m (k D 2; 4; : : : ; 10; 20; 190; 200; m D 10) (˛1 D 10:0; ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

49

CHAPTER

6

Global Sequential Periodic Motions Numerical simulations of stable periodic motions in the global sequential scenario of periodic motions to chaos are carried out through the mid-point integration scheme. The initial conditions and initial time-delay intervals of numerical simulations are obtained from the analytical predictions. The numerical simulations match with the analytical predictions with the accuracy of 10 7 . The harmonic amplitudes and phases are presented for periodic motions. From the frequency-amplitude spectrum, the harmonic terms of periodic motions are presented to provide a clue for approximate analytical solutions to determine how many harmonic terms for a better approximation of periodic motions. The circular symbols are for analytical predictions and the solid curves are for numerical simulation results. The green symbols are for initial time-delay intervals, and the acronyms “D.I.S.” and “D.I.F.” represent delay-initial-start point and delayinitial-ﬁnishing point, respectively. The two paired asymmetric periodic motions are depicted in black and blue circular symbols.

6.1

A SYMMETRIC PERIOD-1 MOTION

In Fig. 6.1, a symmetric period-1 motion is presented at D 5 with an initial condition of x0 D 0:066293. The time-histories of displacement and velocity for the symmetric period-1 motion are presented in Figs. 6.1a and b. The slow-varying zones and spikes are similar to the van der Pol oscillator. The slow-varying zone is shaded by the gray color. The phase trajectory of the symmetric period-1 motion is presented in Fig. 6.1c. The initial time-delay segments are depicted by the green circular symbols. In Fig. 6.1d, the harmonic amplitudes for diﬀerent harmonic orders are presented. For the symmetric period-1 motions, a0 D 0 and A2l D 0 but A2l 1 ¤ 0 (l D 1; 2; 3; : : :). The main harmonic amplitudes are A1 1:1984, A3 0:17437, A5 0:06469, and A7 0:02020. Ak 2 .10 14 ; 10 2 / for k D 9; 11; : : : ; 67 with A67 4:8682e -14. For such a symmetric period-1 motion, four (4) harmonic terms can give an approximate solution with accuracy less than " D 10 2 . For the accuracy of " D 10 14 , 33 odd harmonic terms should be used for such an accurate, symmetric period-1 motion.

50

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

Figure 6.1: A symmetric period-1 motion ( D 5:0): (a) displacement, (b) velocity, (c) trajectory, (d) harmonic amplitude (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

6.2

ASYMMETRIC PERIOD-1 TO PERIOD-8 MOTIONS

The trajectories of paired asymmetric period-1 to period-8 motions are presented in Fig. 6.2. The initial conditions are listed in Table 6.1. The initial time-delayed segment is depicted by green circular symbols. From Eq. (2.1), the time-delayed, nonlinear dynamical system has thee equilibriums (i.e., x1e D 0, x1e D , and x2e D ). Thus, the centers of the asymmetric periodic motions will exist near the equilibrium points of x1e D and x2e D . The center of symmetric periodic motions is at x1e D 0. In Figs. 6.2a and b, trajectories of two paired asymmetric period-1 motions on the left and right sides of x D 0 are presented for D 10:8, respectively. The paired asymmetric period-1 motions are of skew symmetry but the time-delay segments for the right and left period-1 motions lies in the fast and slow varying zones. The trajectory has one cycle on the left or right side of x D 0. From frequency-amplitude analysis, the corresponding constants are a0R D a0L 0:5552, which are the centers of the two paired period-1 motions. The main

6.2. ASYMMETRIC PERIOD-1 TO PERIOD-8 MOTIONS

51

Figure 6.2: Trajectories for G1.A/ . (a) and (b) left and right period-1 motions ( D 10:8). (c) and (d) left and right period-2 motions ( D 10:05). (e) and (f ) left and right period-4 motion ( D 9:95). (g) and (h) left and right period-8 motion ( D 9:947). (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0) (Continues.)

52

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

Figure 6.2: (Continued.) Trajectories for G1.A/ . (a) and (b) left and right period-1 motions ( D 10:8). (c) and (d) left and right period-2 motions ( D 10:05). (e) and (f ) left and right period-4 motion ( D 9:95). (g) and (h) left and right period-8 motion ( D 9:947). (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). Table 6.1: Input data for asymmetric period-1 to period-8 motions in the global scenario of G1.A/ .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/ Figure 6.2 (a) (b) (c) (d) (e) (f ) (g) (h)

Ω 10.8 10.5 9.95 9.947

x0 -0.5368059461525000 0.5735181853464750 -0.6645313199111290 -0.0918276612665433 -0.4708742722512966 -0.2005052178719030 -0.5091086322816001 -0.1929364446579485

Type P-1 (left) P-1 (right) P-2 (left) P-2 (right) P-4 (left) P-4 (right) P-8 (left) P-8 (right)

harmonic amplitudes for the two asymmetric period-1 motions are A1 0:3904, A2 0:0595, and A3 0:0112. Ak 2 .10 14 ; 10 2 / for k D 4; 5; : : : ; 24 with A24 8:001e -14. The harmonic phases of the two paired asymmetric period-1 motions are 'kL D mod.'kR C .k C 1/; 2/ for k D 1; 2; 3; : : :. The primary harmonic term plays an important role on such two asymmetric period-1 motions. With 24 harmonic terms, the analytical solutions of such two asymmetric period-1 motions have the accuracy of " D 10 14 .

6.2. ASYMMETRIC PERIOD-1 TO PERIOD-8 MOTIONS

53

After the period-doubling bifurcation of asymmetric period-1 motion, the corresponding asymmetric period-2 motions are observed in the bifurcation tree. In Figs. 6.2c and d, trajectories of two paired asymmetric period-2 motions on the left and right sides of x D 0 are presented for D 10:05, respectively. The two paired asymmetric period-2 motions are also of skew symmetry. From frequency-amplitude analysis, the corresponding constant terms are a0.2/R D a0.2/L 0:4721 for the center of the two paired period-2 motions. The main harmonic amplitudes for the two asymmetric period-2 motions are A1=2 0:3903, A1 0:2762, A3=2 0:1037; A2 8:4385e -3, and A5=2 0:0117. Ak=2 2 .10 14 ; 10 2 / for k D 6; 7; : : : ; 44 with A22 2:7436e -14. The harmonic phases for the two asymmetric period-2 motions are L R 'k=2 D mod.'k=2 C ..1 C 2r/k=2 C 1/; 2/ for k D 1; 2; 3; : : : and r 2 f0; 1g. For such a numerical simulation, the two harmonic phases are with r D 1, which means the initial conditions are from the point with t0 D rT D T . The dominated harmonic amplitudes are A1=2 0:3903, A1 0:2762, and A3=2 0:1037. The analytical expressions of the two asymmetric period-2 motions need 44 harmonic terms to keep the accuracy of 10 14 . After the period-doubling bifurcation of the asymmetric period-2 motion, the corresponding asymmetric period-4 motion occurs in the same bifurcation tree. Thus, trajectories of two paired asymmetric period-4 motions are presented for D 9:95 in Figs. 6.2e and f. The trajectories of the two period-4 motions double the cycles of the corresponding trajectories of asymmetric period-2 motions. Similarly, the constant terms are a0.4/R D a0.4/L 0:3738 for the two paired period-4 motions. The main harmonic amplitudes for the two asymmetric period-4 motions are A1=4 0:0215, A1=2 0:4653, A3=4 0:0121, A1 0:2144, A5=4 5:1821e -3, A3=2 0:0844, A7=4 4:2122e -3, and A2 0:0251. The other harmonic amplitudes are Ak=4 2 .10 14 ; 10 2 / for k D 6; 7; : : : ; 91 with A91=4 1:6900e -14. The harmonic L R phases for the two asymmetric period-4 motions satisfy 'k=4 D mod.'k=4 C ..1 C 2r/k=4 C 1/; 2/ for k D 1; 2; 3; : : : and r 2 f0; 1; 2; 3g. For such numerical results, the two harmonic phases are with r D 3, which means the initial conditions are from the point with t0 D rT D 3T . The dominated harmonic amplitudes still are A1=2 0:4653, A1 0:2144, and A3=2 0:0844. Such two asymmetric period-4 motions should look like period-2 motions with eﬀects of A1=4 0:0215 and A3=4 0:0121. The analytical expressions of the two period-4 motions need 92 harmonic terms to keep the accuracy of 10 14 . On the bifurcation tree, two paired asymmetric period-8 motions also exist. To illustrate motion variation and complexity, in Figs. 6.2g and h, trajectories of two asymmetric period-8 motions are presented for D 9:947. The trajectories of the two asymmetric period-8 motions double the cycles of the corresponding trajectories of asymmetric period-4 motions. Similarly, the constant terms are a0.8/R D a0.8/L 0:3741 for the two paired period-8 motions, which are just little oﬀ from the center of the asymmetric period-4 motions. The main harmonic amplitudes for the two asymmetric period-8 motions are A1=8 8:6030e -3, A1=4 0:0333, A3=8 8:6706e -3, A1=2 0:4632, A5=8 4:2145e -3, A3=4 0:0184, A7=8 4:5505e -3, A1 0:2163, A9=8 7:4861e -4, A5=4 7:5489e -3, A11=8 1:6132e -3, A3=2 0:0845, A13=8 1:2412e -3,

54

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

Table 6.2: Input data for symmetric period-3 to period-6 motions in the global scenario of G3.S / .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/ Figure 6.3 (a) (b) (c) (d) (e) (f )

Type

Ω

x0

9.5

-0.4371145045940370

P-3 (S)

-0.3333686855017100 0.9439672428736350 -0.3257852977878400 0.9528964687803600

P-3 (A, left) P-3 (A, right) P-6 (A, left) P-6 (A, right)

7.82 7.82754

A7=4 6:3986e -3, A15=8 1:5406e -3, and A2 0:0248. The other harmonic amplitudes are Ak=8 2 .10 14 ; 10 2 / for k D 17; 18; : : : ; 184 with A23 1:4e -14. Harmonic phases for the L R two asymmetric period-8 motions satisfy 'k=8 D mod.'k=8 C ..1 C 2r/k=8 C 1/; 2/ for k D 1; 2; 3; : : :, and r 2 f0; 1; 2; ; 7g. For harmonic phase distributions, the two harmonic phases are with r D 7, which means the initial conditions are from node points with t0 D rT D 7T . The dominated harmonic amplitudes are A1=2 0:4632, A1 0:2163, and A3=2 0:0845 with A1=4 0:0333, and A3=4 0:0184. The main harmonic amplitudes of A1=8 8:6030e -3, A3=8 8:6706e -3, A5=8 4:2145e -3, and A7=8 4:5505e -3 make period-8 motions be diﬀerent from the period-4 motions. The analytical expressions of such two period-8 motions need 184 harmonic terms to keep the accuracy of 10 14 .

6.3

SYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS

In Fig. 6.3, trajectories of symmetric period-3 to period-6 motions are presented with initial conditions in Table 6.2. The trajectory and displacement responses of a symmetric period-3 motion are presented for D 9:5 in Figs. 6.3a and b. The trajectory symmetry of the period-3 motion is observed, and the time-history of displacement is clearly illustrated. For the symmetric period-3 motions, a0.3/ D 0, A2l=3 D 0, and A.2l 1/=3 ¤ 0 for l D 1; 2; 3 : : :. The main harmonic amplitudes are A1=3 0:8709, A1 0:1852, A5=3 0:0739, and A7=3 0:0103. The other harmonic amplitudes are Ak=3 2 .10 14 ; 10 2 / for k D 9; 11; : : : ; 63 with A21 1:2972e -14. For such a symmetric period-3 motion, four (4) harmonic terms can give an approximate solution with the accuracy of " D 10 2 . For the accuracy of " D 10 14 , thirty-one (31) odd harmonic terms should be used for an accurate, symmetric period-3 motion. After the saddle-node bifurcation, the asymmetric period-3 motions appear. In Figs. 6.3c and d, trajectories of two paired, asymmetric period-3 motions are presented for 7:82. Compared to the symmetric period-3 motion, the asymmetric period-3 motions possess a large cycle and a small cycle on both ends, which are from the symme-

6.3. SYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS

55

Figure 6.3: Trajectories for G3.S/ . (a) and (b) symmetric period-3 motion ( D 9:5). (c) and (d) left and right asymmetric period-3 motions ( D 7:82). (e) and (f ) left and right period-6 motion ( D 7:82754). (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0).

56

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

try breaking of symmetric period-3 motion. For the asymmetric period-3 motions, a0.3/ ¤ 0, A2l=3 ¤ 0 and A.2l 1/=3 ¤ 0 for l D 1; 2; 3 : : :. The constants are a0R.3/ D a0L.3/ D 0:0477, which are just a little bit oﬀ from the equilibrium point of x e D 0. However, the centers of the asymmetric periodic motions in asymmetric motion bifurcation trees are close to the equilibriums of x e D ˙. The main harmonic amplitudes are A1=3 0:7297, A2=3 0:2649, A1 0:4547, A4=3 0:0518, A5=3 0:0503, A2 0:0815, A7=3 0:0375, A8=3 5:0736e -3, A3 0:0131, and A10=3 0:0151. The other harmonic amplitudes are Ak=3 2 .10 14 ; 10 2 / for k D 11; 12; : : : ; 103 with A103=3 1:2868e -14. The most important harmonic amplitudes are A1=3 0:7297, A2=3 0:2649, and A1 0:4547. To keep the accuracy of " D 10 14 , 103 harmonic terms should be considered. The harmonic phases for the two asymmetric periodL R D mod.'k=3 C ..1 C 2r/k=3 C 1/; 2/ for k D 1; 2; : : : with 3 motions satisfy a relation 'k=3 r D 2. For the period-doubling bifurcation of such asymmetric period-3 motions, asymmetric period-6 motions will appear. In Figs. 6.3e and f, trajectories of two paired, asymmetric period-6 motions are presented for 7:82754. The asymmetric period-6 motions are very close to the asymmetric period-3 motions. The constants are a0R.6/ D a0L.6/ D 0:0168. The main harmonic amplitudes are A1=6 5:8685e -4, A1=3 0:6010, A1=2 6:040e -4, A2=3 0:3701, A5=6 5:0340e -4, A1 0:4063, A7=6 4:2544e -4, A4=3 0:0614, A3=2 3:1263e -4, A5=3 0:0238, A11=6 2:4424e -4, A2 0:0912, A13=6 1:9234e -4, A7=3 0:0147, A1=6 1:2866e -4, A8=3 6:7303e -3, A17=6 9:7112e -5, A3 0:01925, A19=6 6:5977e -5, and A10=3 0:0107. Ak=6 2 .10 14 ; 10 2 / for k D 21; 22; : : : ; 208 with A104=3 1:030e -14. The harmonic terms eﬀecting on the period-6 motions possess very small values. Thus, such asymmetric period-6 motions looks like the corresponding asymmetric period-3 motions. The most important harmonic amplitudes are A1=3 0:6010, A2=3 0:3701, and A1 0:4063. To keep the accuracy of " D 10 14 , 208 harmonic terms should be considered. The harmonic phases for L R the two asymmetric period-6 motions are with 'k=6 D mod.'k=6 C ..1 C 2r/k=6 C 1/; 2/ for k D 1; 2; : : : with r D 0.

6.4

ASYMMETRIC PERIOD-2 TO PERIOD-8 MOTIONS

In Figs. 6.4a and b, trajectories of two asymmetric period-2 motions on the left and right sides of x D 0 are presented for D 7:171 in the global scenario of G2.A/ , respectively. The initial conditions are tabled in Table 6.3. Such trajectories of the asymmetric period-2 motion includes all equilibriums. The constant terms are a0.2/R D a0.2/L 0:2541. The main harmonic amplitudes for the two paired asymmetric period-2 motions are A1=2 0:7095, A1 0:5532, A3=2 0:0757, A2 0:0864, A5=2 0:0669, A3 0:0200, A7=2 0:0137, and A3 0:0142. Ak=2 2 .10 14 ; 10 2 / for k D 9; 10; : : : ; 81 with A81=2 1:3672e -14. The harmonic phases L R for the two asymmetric period-2 motions satisfy a relationship as 'k=2 D mod.'k=2 C ..1 C 2r/k=2 C 1/; 2/ for k D 1; 2; 3; : : : and r D 0. The dominated harmonic amplitudes are

6.4. ASYMMETRIC PERIOD-2 TO PERIOD-8 MOTIONS

57

Figure 6.4: Trajectories forG2.A/ . (a) and (b) left and right period-2 motions ( D 7:171). (c) and (d) left and right period-4 motions ( D 7:021). (e) and (f ) left and right period-8 motion ( D 7:012). (˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0).

58

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

Table 6.3: Input data for asymmetric period-2 to period-8 motions in the global scenario of G2.A/ .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/ Figure 6.4 (a) (b) (c) (d) (e) (f )

Ω 7.171 7.021 7.012

x0 -0.2756040180625460 -0.5589843311776160 -0.3897313787763300 -0.6141686236437800 -0.4370828654587291 -0.6523689627768331

Type P-2 (A, left) P-2 (A, right) P-4 (A, left) P-4 (A, right) P-8 (A, left) P-8 (A, right)

A1=2 0:7095 and A1 0:5532. The analytical expressions of the period-2 motions needs 81 harmonic terms to keep the accuracy of 10 14 . Such asymmetric period-2 motions are diﬀerent from the asymmetric period-2 motions on the bifurcation tree of the asymmetric period-1 motion. On the same bifurcation tree of the asymmetric period-2 motion, trajectories of the asymmetric period-4 motions are presented for D 7:021 in Figs. 6.4c and d. The trajectories of the two asymmetric period-4 motions double the corresponding trajectories of the asymmetric period-2 motions. The corresponding constant terms are a0.4/R D a0.4/L 0:2334 for the two paired period-4 motions. The main harmonic amplitudes for the two asymmetric period-4 motions are A1=4 0:0663, A1=2 0:6481, A3=4 0:1621, A1 0:5863, A5=4 0:0420, A3=2 0:0385, A7=4 0:0390, A2 0:0524, A9=4 0:0534, A5=2 0:0702, A11=4 5:1686e -3, A3 0:0183, A13=4 9:9641e -3, A7=2 4:7657e -3, A15=4 0:0115, and A4 0:0122. The other harmonic amplitudes are Ak=4 2 .10 14 ; 10 2 / for k D 17; 18; : : : ; 160 with A40 1:5365e -14. L R The harmonic phases for the two asymmetric period-4 motions are with 'k=4 D mod.'k=4 C ..1 C 2r/k=4 C 1/; 2/ for k D 1; 2; 3; : : : with r D 0. The dominated harmonic amplitudes are A1=2 0:6481, A3=4 0:1621, and A1 0:5863. The analytical expressions of the two period-4 motions need 160 harmonic terms to keep the accuracy of 10 14 . Such asymmetric period-4 motions in the bifurcation tree of asymmetric period-2 motions are diﬀerent from the asymmetric period-4 motions on the bifurcation tree of the asymmetric period-1 motions to chaos. On the same bifurcation tree, two paired asymmetric period-8 motions are presented in Figs. 6.4e and f for D 7:012. The trajectories of the two asymmetric period8 motions double the corresponding trajectories of asymmetric period-4 motions. The corresponding constant terms are a0.8/R D a0.8/L 0:2306 for the two paired period8 motions, which are just little oﬀ the corresponding centers of the period-4 motions. The main harmonic amplitudes for the two asymmetric period-8 motions are A1=8

6.5. SYMMETRIC PERIOD-5 TO PERIOD-10 MOTIONS

59

0:0103, A1=4 0:0691, A3=8 5:3612e -3, A1=2 0:6462, A5=8 0:0327, A3=4 0:1821, A7=8 0:0462, A1 0:5760, A9=8 0:0182, A5=4 0:0432, A11=8 0:0158, A3=2 0:0301, A13=8 9:5852e -3, A7=4 0:0363, A15=8 3:8780e -3, A2 0:0490, A17=8 6:4642e -3, A9=4 0:0562, A19=8 0:0131, A5=2 0:0661, A21=8 0:0106, A11=4 4:4962e -3, A23=8 6:0716e -4, A3 0:0192, A25=8 3:3765e -3, A13=4 8:5028e -3, A27=8 5:3750e -4, A7=2 4:2734e -3, A29=8 1:4523e -3, A15=4 0:0115, A31=8 2:8020e -3, and A4 0:0105. The other harmonic amplitudes are Ak=8 2 .10 14 ; 10 2 / for k D 33; 34; : : : ; 328 with A41 L 1:3e -14. The harmonic phases for the two asymmetric period-8 motions satisfy 'k=8 D R mod.'k=8 C ..1 C 2r/k=8 C 1/; 2/ for k D 1; 2; 3; : : : and r D 0. The dominated harmonic amplitudes are still A1=2 0:6462, A3=4 0:1821, and A1 0:5760 for the period-8 motion. The analytical expressions of such two period-8 motions need 328 harmonic terms to keep the accuracy of 10 14 . Such asymmetric period-8 motions in the bifurcation tree of asymmetric period-2 motions are diﬀerent from the asymmetric period-8 motions on the bifurcation tree of the asymmetric period-1 motions to chaos.

6.5

SYMMETRIC PERIOD-5 TO PERIOD-10 MOTIONS

In Fig. 6.5, trajectories of symmetric period-5 to period-10 motions are presented with initial conditions in Table 6.4. The trajectory and displacement responses of a symmetric period5 motion are presented for D 6:9 in Figs. 6.5a and b. The trajectory symmetry of the period-5 motion is observed, and the time-history of displacement is also illustrated. For the symmetric period-5 motions, a0.5/ D 0, A2l=5 D 0, and A.2l 1/=5 ¤ 0 for l D 1; 2; 3 : : :. The main harmonic amplitudes are A1=5 0:4065, A3=5 0:7603, A1 0:1852, A7=5 0:0727, A9=5 0:0300, A11=5 0:0743, A13=5 0:0475, A3 0:0260, A17=5 4:0767e -3, A19=5 0:0103, and A21=5 0:0125. The other harmonic amplitudes are Ak=5 2 .10 14 ; 10 2 / for k D 23; 25; : : : ; 213 with A213=5 1:23e -14. For such a symmetric period-5 motion, eleven (11) harmonic terms can give an approximate solution with the accuracy of " D 10 2 . For the accuracy of " D 10 14 , 106 odd harmonic terms should be used for an accurate, symmetric period-5 motion. In Figs. 6.5c and d, trajectories of two paired, asymmetric period-5 motions are presented for 6:632. Compared to the symmetric period-5 motion, the asymmetric period-5 motion possesses two similar cycles on the left and right ends. For the asymmetric period-5 motions, the constants are a0R.5/ D a0L.5/ D 5:4447e -3, which are just a little bit oﬀ from the equilibrium point of x e D 0. The main harmonic amplitudes are A1=5 0:3931, A2=5 0:0147, A3=5 0:5877, A4=5 0:0289, A1 0:6810, A6=5 0:0103, A7=5 0:0469, A8=5 8:9125e -3, A9=5 0:0552, A2 9:6829e -3, A11=5 0:0435, A12=5 0:0103, A13=5 0:0948, A14=5 2:1960e -3, : : :, A3 0:0196, A21=5 0:0205, A22=5 1:2636e -3, and A23=5 0:0112. Ak=5 2 .10 14 ; 10 2 / for k D 24; 25; : : : ; 228 with A228=5 1:02e -14. The most important harmonic amplitudes are A1=5 0:3931, A3=5 0:5877, and A1 0:6810. To keep the accuracy of " D 10 14 , 228 harmonic terms should be considered. The harmonic phases

60

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

Figure 6.5: Trajectories for G5.S/ . (a) and (b) symmetric period-5 motion plus displacement ( D 6:9). (c) and (d) left and right asymmetric period-5 motions ( D 6:632). (e) and (f ) left and right period-10 motion ( D 6:61). (˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0).

6.6. ASYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS

61

Table 6.4: Input data for symmetric period-5 to period-10 motions in the global scenario of G5.S/ .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/ Figure 6.5 (a) (b) (c) (d) (e) (f )

Type

Ω

x0

6.9

-0.5765726010903134

P-5 (S)

-0.7747098333268110 0.8501225096456310 -0.4998561879479400 -0.2927316039513600

P-5 (A, left) P-5 (A, right) P-10 (A, left) P-10 (A, right)

6.632 6.61

L R for the two asymmetric period-5 motions satisfy a relation 'k=5 D mod.'k=5 C ..1 C 2r/k=5 C 1/; 2/ for k D 1; 2; : : : with r D 0. On the bifurcation tree of the symmetric period-5 motion, the trajectories of a paired, asymmetric period-10 motions are presented for 6:61 in Figs. 6.5e and f. The asymmetric period-10 motions double the corresponding trajectories of the asymmetric period-5 motions. The corresponding constants in the ﬁnite Fourier series are a0R.10/ D a0L.10/ D 0:0198. The main harmonic amplitudes are A1=10 4:4782e -3, A1=5 0:3887, A3=10 0:0115, A2=5 0:0577, A1=2 8:7197e -3, A3=5 0:5870, A7=10 0:0244, A4=5 0:1248, A9=10 0:0353, and A1 0:6608. Ak=10 2 .10 14 ; 10 1 / for k D 11; 12; : : : ; 442 with A442=10 1:18e -14. The most important harmonic amplitudes are A1=5 0:3887, A3=5 0:5870, A4=5 0:1248, and A1 0:6608. To keep the accuracy of " D 10 14 , 442 harmonic terms should be considered. L The harmonic phases for the two asymmetric period-10 motions satisfy a relation 'k=10 D R mod.'k=10 C ..1 C 2r/k=10 C 1/; 2/ for k D 1; 2; : : : with r D 9.

6.6

ASYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS

In Figs. 6.6a and b, the trajectories of the paired, asymmetric period-3 motions are presented for 6:5. The constants are a0R.3/ D a0L.3/ D 0:1693. The main harmonic amplitudes are A1=3 0:4082, A2=3 0:6711, and A1 0:5759. Ak=3 2 .10 14 ; 10 1 / for k D 4; 5; : : : ; 139 with A139=3 1:36e -14. To keep the accuracy of " D 10 14 , 139 harmonic terms should be considered. The harmonic phases for the two asymmetric period-3 motions satisfy a relation L L 'k=3 D mod.'k=3 C ..1 C 2r/k=3 C 1/; 2/ for k D 1; 2; : : : with r D 2. Such asymmetric period-3 motions in the bifurcation tree of asymmetric period-3 motions are diﬀerent from the asymmetric period-3 motions on the bifurcation tree of the symmetric period-3 motions to chaos. In Figs. 6.6c and d, the trajectories of a paired, asymmetric period-6 motions are presented for 7:82754. The constants are a0R.3/ D a0L.3/ D 0:1541. The main harmonic am-

62

6. GLOBAL SEQUENTIAL PERIODIC MOTIONS

Figure 6.6: Trajectories for G3.A/ . (a) and (b) left and right asymmetric period-3 motions ( D 6:5). (c) and (d) left and right period-6 motion ( D 6:373). (˛1 D 10:0, ˛2 D 5:0, ˇ D 10:0, Q0 D 3:0). plitudes are A1=6 0:0320. A1=3 0:3535, A1=2 0:0488, A2=3 0:5402, A5=6 0:0904, and A1 0:7193. Ak=6 2 .10 14 ; 10 2 / for k D 7; 8; : : : ; 282 with A47 1:68e -14. The harmonic terms eﬀecting period-6 motions possess very small values. Thus, such asymmetric period-6 motions look like the corresponding asymmetric period-3 motions. The most important harmonic amplitudes are A1=3 0:3535, A2=3 0:5402, and A1 0:7193. To keep the accuracy of " D 10 14 , 288 harmonic terms should be considered. The harmonic phases for the right and left L L asymmetric period-6 motions are 'k=6 D mod.'k=6 C ..1 C 2r/k=6 C 1/; 2/ for k D 1; 2; : : : with r D 5. Such asymmetric period-6 motions in the bifurcation tree of asymmetric period3 motions are diﬀerent from the asymmetric period-6 motions on the bifurcation tree of the symmetric period-3 motions to chaos. Similarly, trajectories of other periodic motions in the global sequential scenario of bifurcation trees can be illustrated.

6.6. ASYMMETRIC PERIOD-3 TO PERIOD-6 MOTIONS

63

Table 6.5: Input data for asymmetric period-3 to period-6 motions in the global scenario of G3.A/ .˛1 D 10:0; ˛2 D 5:0; ˇ D 10:0; Q0 D 3:0/ Figure 6.6 (a) (b) (c) (d)

Ω 6.5 6.373

x0 -0.5757096061403786 -0.4353400464079145 -0.1589424200076042 0.9634305600268458

Type P-5 (A, left) P-5 (A, right) P-10 (A, left) P-10 (A, right)

65

CHAPTER

7

Conclusive Remarks In this book, the global sequential scenario of bifurcation trees of periodic motions to chaos is presented through a 1-dimensional, time-delayed, nonlinear dynamical system. The sequential periodic motions in such a 1-dimensional time-delayed system is achieved semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential order of bifurcation trees of periodic motions to chaos is given by G1.S/ G G1.A/ G G3.S/ G G2.A/ G Gm.A/ G G2mC1.S/ G .m D 1; 2; : : :/;

where Gm.A/ represents a global bifurcation tree of an asymmetric period-m motion to chaos and G2mC1.S/ is for a global bifurcation tree of a symmetric period-.2m C 1/ motion to chaos. Each bifurcation tree of a speciﬁc periodic motion to chaos are presented in detail. The bifurcation tree appearance and vanishing are determined by the saddle-node bifurcation, and the cascaded period-doubled periodic solutions are determined by the period-doubling bifurcation. From ﬁnite Fourier series, harmonic amplitude and harmonic phases for periodic motions on the global bifurcation tree are obtained for frequency analysis. Numerical illustrations of periodic motions are given for complex periodic motions in global bifurcation trees. The rich dynamics of the 1-D, delayed, nonlinear dynamical system is presented. Such global sequential periodic motions to chaos also exist in nonlinear dynamical systems.

67

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Authors’ Biographies SIYUAN XING Dr. Xing is an assistant professor at California Polytechnic State University. He received a B.S. from Sichuan University in 2013, an M.S. from Southern Illinois University Edwardsville in 2016, and a Ph.D. from Southern Illinois University Carbondale, in 2019. His research interests are in the area of nonlinear dynamics and time-delay systems. Dr. Xing has published 3 book chapters, 13 peer-review journal papers, and 8 conference papers on nonlinear dynamics.

ALBERT C.J. LUO Professor Luo works at Southern Illinois University Edwardsville. For over 30 years, Dr. Luo’s contributions on nonlinear dynamical systems and mechanics lie in (i) the local singularity theory for discontinuous dynamical systems, (ii) dynamical systems synchronization, (iii) analytical solutions of periodic and chaotic motions in nonlinear dynamical systems, (iv) the theory for stochastic and resonant layer in nonlinear Hamiltonian systems, and (v) the full nonlinear theory for a deformable body. Such contributions have been scattered into 20 monographs and over 300 peer-reviewed journal and conference papers. Dr. Luo served as an editor for the journal Communications in Nonlinear Science and Numerical simulation, and book series on Nonlinear Physical Science (HEP) and Nonlinear Systems and Complexity (Springer). Dr. Luo was an editorial member at IMeChE Part K, Journal of Multibody Dynamics, and Journal of Vibration and Control), and has organized over 30 international symposiums and conferences on Dynamics and Control.