Dynamics and bifurcations [Corrected] 0387971416, 9780387971414

Table of contents :
Cover
Title page
Greeting
Contents
PART I: Dimension One
Chapter 1. Scalar Autonomous Equations
1.1. Existence and Uniqueness
1.2. Geometry of Flows
1.3. Stability of Equilibria
1.4. Equations on a Circle
Chapter 2. Elementary Bifurcations
2.1. Dependence on Parameters - Examples
2.2. The Implicit Function Theorem
2.3. Local Perturbations Near Equilibria
2.4. An Example on a Circle
2.5. Computing Bifurcation Diagrams
2.6. Equivalence of Flows
Chapter 3. Scalar Maps
3.1. Euler's Algorithm and Maps
3.2. Geometry of Scalar Maps
3.3. Bifurcations of Monotone Maps
3.4. Period-doubling Bifurcation
3.5. An Example: The Logistic Map
PART II: Dimension One and One Half
Chapter 4. Scalar Nonautonomous Equations
4.1. General Properties of Solutions
4.2. Geometry of Periodic Equations
4.3. Periodic Equations on a Cylinder
4.4. Examples of Periodic Equations
4.5. Stability of Periodic Solutions
Chapter 5. Bifurcation of Periodic Equations
5.1. Bifurcations of Poincar\303\251 Maps
5.2. Stability of Nonhyperbolic Periodic Solutions
5.3. Perturbations of Vector Fields
Chapter 6. On Tori and Circles
6.1. Differential Equations on a Torus
6.2. Rotation Number
6.3. An Example: The Standard Circle Map
PART III: Dimension Two
Chapter 7. Planar Autonomous Systems
7.1. \"Natural\" Examples of Planar Systems
7.2. General Properties and Geometry
7.3. Product Systems
7.4. First Integrals and Conservatjve Systems
7.5. Examples of Elementary Bifurcations
Chapter 8. Linear Systems
8.1. Properties of Solutions of Linear Systems
8.2. Reduction to Canonical Forms
8.3. Qualitative Equivalence in Linear Systems
8.4. Bifurcations in Linear Systems
8.5. Nonhomogeneous Linear Systems
8.6. Linear Systems with 1-periodic Coefficients
Chapter 9. Near Equilibria
9.1. Asymptotic Stability from Linearization
9.2. Instability from Linearization
9.3. Liapunov Functions
9.4. An Invariance Principle
9.5. Preservation of a Saddle
9.6. Flow Equivalence Near Hyperbolic Equilibria
9.7. Saddle Connections
Chapter 10. In the Presence of a Zero Eigenvalue
10.1. Stability
10.2. Bifurcations
10.3. Center Manifolds
Chapter 11. In the Presence of Purely Imaginary Eigenvalues
11.1. Stability
11.2. Poincar\303\251-Andronov-Hopf Bifurcation
11.3. Computing Bifurcation Curves
Chapter 12. Periodic Orbits
12.1. Poincar\303\251-Bendixson Theorem
12.2. Stability of Periodic Orbits
12.3. Local Bifurcations of Periodic Orbits
12.4. A Homoclinic Bifurcation
Chapter 13. All Planar Things Considered
13.1. Structurally Stable Vector Fields
13.2. Dissipative Systems
13.3. One-parameter Generic Bifurcations
13.4. Bifurcations in the Presence of Symmetry
13.5. Local Two-parameter Bifurcations
Chapter 14. Conservative and Gradient Systems
14.1. Second-order Conservative Systems
14.2. Bifurcations in Conservative Systems
14.3. Gradient Vector Fields
Chapter 15. Planar Maps
15.1. Linear Maps
15.2. Near Fixed Points
15.3. Numerical Algorithms and Maps
15.4. Saddle Node and Period Doubling
15.5. Poincar\303\251-Andronov-Hopf Bifurcation
15.6. Area-preserving Maps
PART IV: Higher Dimensions
Chapter 16. Dimension Two and One Half
16.1. Forced Van der Pol
16.2. Forced Duffing
16.3. Near a Transversal Homoclinic Point
16.4. Forced and Damped Duffing
Chapter 17. Dimension Three
17.1. Period Doubling
17.2. Bifurcation to Invariant Torus
17.3. Silnikov Orbits
17.4. The Lorenz Equations
Chapter 18. Dimension Four
18.1. Integrable Hamiltonians
18.2. A Nonintegrable Hamiltonian
FAREWELL
APPENDIX: A Catalogue of Fundamental Theorems
REFERENCES
INDEX

Citation preview

K. Hale

Jack

Bifurcations)

314

Ko\nak)

and

Dynamics

With

Htiseyin

Illustrations)

Springer)))

H iiseyin

K. Hale

Jack

School of

Mathematics

Institute

Georgia

Ko\037ak

of Mathematics

Department

of Technology

Atlanta, GA 30332 USA

of Miami

University

Coral

and

Science

Computer

FL

Gables,

33124

USA

[email protected])

[email protected]) Editors

L.

Marsden

IE.

Control and Dynamical Systems, Institute of Technology California

Brown

CA 91125

Pasadena,

Sirovich

of Applied

Division

104-44

Mathematics

University

RI 02912

Providence,

USA

USA

M. Golubitsky

W.

of Mathematics

Department

University of Houston Houston, TX 77004

Universitat

USA)

and text art

Cover

Mathematics

Subject

Halil

by

Jager

of Congress

Catalog

Printed

on acid-free

paper.)

Heidelberg

1m Neuenheimer

Feld 294

6900

FRG)

Heidelberg,

Buttann.)

58 Fxx, 34 xx, 58 F

Classifications:

Library

of Applied Mathematics

Department

Card

Number:

14)

92-10512)

New York, Inc. This work may not be translated or copied in whole or in part without the 175 Fifth Avenue, New York, NY 10010, written (Springer-Verlag, permission of the publisher Use in reviews or scholarly analysis. for brief excerpts in connection with USA), except electronic comconnection with any form of information storage and retrieval, adaptation, is now known or hereafter or by similar or dissimilar developed software, methodology puter forbidden. even The use of general etc., in this publication, names, trade names, trademarks, descriptive as a sign that such names, as is not to be taken if the former are not especially identified, Marks and Merchandise understood Act, may accordingly be used freely by by the Trade

@ 1991

All

rights

Springer-Verlag reserved.

anyone.) from the authors' TEX file. Photocomposed copy prepared & Sons, Harrisonburg, Printed and bound by R.R. Donnelley in the United States of America.) Printed

9 87 6 5 4 3

(Corrected

third

printing,

Virginia.

1996))

ISBN

0-387-97141-6

Springer-Verlag

New

ISBN

3-540-97141-6

Springer- Verlag

Berlin

York

Berlin

Heidelberg

Heidelberg New York

SPIN

10549030)))

To Who

are the for

Students:)

primary the ou

reason

of

existence r profession

and

this

book)))

Greeting)

Thank

you

and

it is

As

us

the

about

an unusual

explain

it

how

in

book

evolved

will

find ideas

and both

difference

content

at

Brown

bifur-

equations.

and style, let

courses

in

the

Dividuring

University

being.

and

alias equations, dynamical in science, one which chapter germias celestial fields such mechanics, nonlinear oscillations, Over the centuries, as a result of the efforts of sciendynamics. mathematicians alike, an attractive and far-reaching theory has

systems, is an old in applied nated

and fluid tists and

into

you

of dynamics and

from our

sion of Applied Mathematics a three-year period, and came The subject of differential

Inside

geometry

differential

of ordinary

cations

our book.

for opening

examples

and

difference

much-honored

of computers, years, due primarily to the proliferation has once more in turned to its roots with dynamical systems applications a more mature outlook. Currently, the level of excitement and perhaps front but in almost all allied fields activity, not only on the mathematical It is the aim of our book to provide a modest founof learning, is unique. of these dation for taking facets part in certain theoretical and practical exciting developments. of dynamical The subject accessible to systems is a vast one not easily and beginning graduate students in mathematics or science undergraduate and engineering. Many of the available books and expository narratives either extensive mathematical or are not designed to require preparation, be used as textbooks. It is with the desire to fill this void that we have In

emerged.

written

recent

the present

It is both damental ple

setting,

our

book. conviction

and

our

ideas of dynamics and bifurcations one that is mathematically

experience can insightful

that many be

explained

yet

devoid

of

the

fun-

in a simof

extensive)))

viii

Greeting)

we have opted in the present book to proceed Accordingly, by low-dimensional dynamical systems. We will momentarily give a brief of some of the central topics of our book, one which necessarily summary If you are a beginning student of dynamterms. contains some technical rest assured that precise mathematical definitions of all these ics, however, as as realizations the well of in specific terms, ample phenomena dynamical will unfold as turn the equations, you pages. in dimensions one, \"one and one half,\" and two constitute Equations the majority of the text. are devoted Indeed, nearly one hundred pages to scalar equations where, their and despite simplicity apparent triviality, ideas of our subject are already visible. We many of the contemporary in particular, that the basic notions of stability and bifurcademonstrate, tions of vector fields are easily explained for scalar autonomous equationsdimension one-because their are determined flows from the equilibrium formalism.

We

points.

to

scalar

exciting,

that may

doubling

bifurcation,

furcations

how

also explore and show

of periodic

chaos,

etc.

solutions of

approximation

We then

turn to

nonautonomous

equations

the

lead

and is poor-periodand bidynamics

albeit

\"anomalies,\"

numerical

when

arise

of such

solutions

numerical

some of the

maps,

equations

profound

with

periodic

coefficients-dimension one and one half-where scalar natmaps reappear In our discussion of the stability of periodic somaps. urally as Poincare lutions of such equations, we demonstrate how one naturally encounters ideas from the transformation elementary but essential theory of differential equations-normal form theory. These ideas, in the context presented of scalar equations, and more importantly, the philosophical outlook of the recur in later chapters, with a that these ideas convey, subject frequently few

technical

embellishments.

proceed to

autonomous new equations-dimension two--where, equilibria, dynamical such as periodic and homoclinic orbits, appears. In studying behavior, the stability of an equilibrium point, we touch certain subtle topoupon of linear systems as well as the standard aspects logical theory of Liapunov functions. The bifurcation of equilibriurn points of planar equations theory rise to a number of new ideas. for example, the bifurcation gives When, is to other one is led naturally to introduce center manifolds equilibria, and the method of Liapunov-Schmidt to make a reduction to a scalar autonomous equation. The other bifurcation from an equilibrium important is to a periodic orbit-Poincare-Andronov-Hopf bifurcation-and point its analysis can be reduced to that of a nonautonomous periodic equation. There are, of course, other properties of planar differential that equations are more global in character and hence be investigated in terms cannot of scalar chosen to equations. Among these interesting topics, we have include the Poincare-Bendixson theory of planar limit sets, geometry and of conservative bifurcations and gradient systems, and a discussion of struc-))) We next

the

investigate

in

dynamics addition to

of planar

ix)

Greeting

tural

of course, an

stability-with,

include an

We subsequently

of the

theory

ideas

the

on

emphasis

than on

rather

details.

technical

extensive

of

maps.

planar

of certain

discussion

abbreviated

As in

the case of with numerical

scalar

aspects we explore

equations,

difficulties associated solutions of differential in of the and of such bifurcations equations light dynamics maps. To indicate not the richness but also the of this topic, only bewildering complexity we include computer simulations of some of the famous maps. The final part of the book consists of several substantial in examples This section is more disdimensions \"two and one half,\" three, and four. cursive than the previous it is more like a preview to provide ones; designed a smooth entry into certain areas of current research-forced oscillations, of

some

the

strange attractors, of

invited to browse the entries, however, of mathematics, parts

most

tral of

list of the

course,

ruse

must

one

place,

while

general

face the task

especially

in applications.

most

abstractly inclined, grappling irreplaceable source of general in mind, the text and the philosophy

to

with

specific theoretical

be an

numerous

interest.

practical

systems, etc. book, you are,

of

the

analyzing

Moreover,

out

to

equations

the

for

of theoretical and

the dynamics of unraveling insurmountable. The analytically

be

dynanlics

even

examples usually proves observations. With this alike are interwoven with

Unfortunately,

often turns

tions

exercises

difference

and

differential

specific

the

Table of Contents. As you pethat in dynamical systems, as in theorems certainly occupy a cen-

ultimately

equations,

specific

the

in mind

in

covered

topics

through bear

Hamiltonian

integrable

completely

chaos,

detailed

more

a

For

equa-

specific

computer, in

its utility in this is, however, beginning to prove our favorite computer program is, of course, An Animator/Simulator PHASER: for Dynamical Systems, accompanyand Difference Equations through ing one of our earlier books Differential students Our have found PHASER to be an ideal Computer Experiments. in dynamical medium to see the \"dynamics\" and to do some of systems of the it to produce their assignments; we, too, used illustrations for many all

its

versatility,

present

pursuit. On

our

that

front,

book.

Dynamical f3

new

this

Bifurcations

systems

you will be inclined our other favorite

course,

We contributed

would

is

arouse

will

to

like

unselfishly

area. We hope that in interest bifurcation your theory to explore this exciting subject further and vibrant

a vast

book-Methods

record

to the

realization

of our

of

our

book.

of

using,

Theory.

of Bifurcation

in closing our gratitude

Dynamics sufficiently

to

those

have

who

In particular,

the

students-a

lively group consisting of unand graduate students of pure and applied and mathematics, dergraduate of science and engineering-helped our ideas and considerably in fixing Critical readings of the setting realistic bounds for our own enthusiasm.

enthusiastic

text

participation

and insightful

suggestions

by

Nathaniel

Chafee,

Brian

Coomes,

Philip)))

x

Greeting)

Davis,

Robert

Horta,

\037ahin

Ko.. is a small scalar

3

in Eq.

term

ds

==

The

0,

from

(5.8),

second-order

parameter,

we

can

the differential

1

determine

equation)

the function

,

(t

),

leC/'O)

equation jj + y3 - 2>\"y + y = 0, where - theory of sound and is known))) /1,) (5.10))))

m(t) the

arises == in

1)

Scalar

Autonomous

Equations)

IIiiIiiiiiiIi I)))

we present selected basic conchapter, opening about the geometry of solutions of ordinary differTo keep the ideas free from ential technical equations. the setting is one-dimensional-the scalar complications, this

In

cepts

I

these

ity,

pear in various

tion

of

we

examples,

To

ometrically.

as vector

conclude

facilitate

a theorem on what

explain

the

Following existence

geometric

simplic-

and

subject

a differential

analysis,

qualitative

our

and

reap-

a collecunique-

equation is concepts

ge-

such

in and limit set are included of stability of an equilibin determining stability. the role of linear approximation of a scalar the chapter with an example differential equacircle.) on a one-dimensional space other than the realline--a

field,

this discussion. rium point and

tion defined

we

to book.

the

throughout

state

first

Then

Despite their

equations.

are central

concepts

incarnations

ness of solutions.

We

differential

autonomous

i \037

orbit,

point,

equilibrium

The next

topic

is

the

notion

4

this

their

our notation

we establish

section

introductory

tions and

Equations)

and Uniqueness

Existence

1.1. In

Autonomous

Scalar

1:

Chapter

solutions.

Then,

for

differential

motivational

several

after

existence and uniqueness theorem. be an open interval of the real line IR and

equa-

examples,

I

Let

x : I ---+ IR;

t

\037

let)

x(t))

real-valued differentiable function of a real variable t. We will x to denote the derivative dx / dt, and refer to t as time variable. let) Also, independent

be a

use

the

or

the

notation

\037

f(x))

real-valued function. In Chapter 1, we

a given

equations

x

: IR ---+ IR;

f

be

we

a basic

state

of the

will

differential

consider

form)

x == f(x),)

(1.1))

x is an unknown function of t and f is a given function of x. Equation (1.1) is called a scalar autonomous differential equation; scalar because x is one dimensional (real-valued) and autonomous because the function f does not on t. depend I if interval We say that a function x is a solution of Eq. (1.1) on the == I. in a solution for all t will often be We interested E specific x(t) f(x(t)) of Eq. (1.1) which at some initial time we to E I has the value Xo. Thus will study x satisfying) where

x == f(x),)

Equation the

to as an

(1.2) is referred

solutions is

a

called

loss

of

through

of the

character

autonomous

there is no

solution

Xo.)

(1.2))

and problem any of its A useful consequence of in Eq. (1.2) is that equation

initial-value

Xo at to.

differential

in assuming

generality

==

x(to)

that the

initial

value-problem

is

with to == 0, and we will often tacitly do so. To wit, let x(t) be a solution of Eq. (1.2) through Xo at to and define y(t) = x(t + to). Now, observe that y(t) is a solution of Eq. (1.2) through Xo at zero since) specified

y(t) As you

through variables,\"

==

x(t

+ to)

may recall

Xo at to is by the

==

+ to))

f(x(t

from

your

given

implicitly,

==

and

f(y(t))

y(O)

==

Xo.)

a solution of Eq. studies, of \"separation the method

previous using

(1.2) of

formula)

x l

1

s XQ f ( )

ds =

t-

to,)

(1.3))))

1.1. Existence

function

the

One

is defined. integral on the left-hand

the

when

use this formula

to

It is

to

important

in this book is

5)

Uniqueness

the inverse of Occasionally, we will

finding

by

x(t)

this

equation.

of special

solutions

exhibit

purposes of illustrations. However, and one should these integrations solutions.

obtains

of

side

and

differential

for the

equations

it is impossible to perform general, not expect to obtain explicit formulas for this fact from the beginning. In fact, in

realize

to understand as

much as possible about the objective without the knowledge of an behavior of solutions of differential equations formula for the solutions. explicit and their Let us now equations give several examples of differential to realize some of the difficulties that arise in laying the solutions in order and the uniqueness of for the theory, that is, the existence foundations solutions of Eq. (1.2).

our

Example 1.1. The

the differential

Consider

example:

first

equation)

x == -x.)

It

can

be seen

through

at

Xo

it is

0, and

==

defined

only solution of Eq.

(1.4) satisfying 1.2. Finite time: Consider

Example

x

to

It is easy

== x

direct

by

verify

that

differentiation

simple

by

to

(1.4

for

the

2 x(O)

,)

substitution,

is a

e-txo

value

Question: x(O) == xo?

initial-value

problem)

initial

the

==

x(t)

t E IR.

all

==

is

))

solution this

the

I)

Xo.)

(1.5))

or using

formula

that

(1.3),

the

function) xo x(t) = 1 xot

solution.

is a \"nice,\"

the

Notice

solution

x(t)

the function

although

that,

is defined

on the

interval

2

is remarkably for Xo > 0, Xo < O. The importance on all of IR and defined

f(x)

(-00,

==

x

1/xo)

on (-00, +00) for Xo == 0, and (1/xo, +00) for is that the solution is not always of this example of definition of the solution varies with the initial the interval as t approaches the solution becomes unbounded Furthermore, interval of definition. of the I) boundary Example

1.3.

x == IX,)

A

solution

the problem

is given by which

solution

above

does

Consider

solutions:

Multiple

x(t)

==

x(O)

==

is identically

not have

(t +

xo,)

the initial-value with

x >

condition.

1/xo,

the

problem)

O.)

== 0, then there is also 2VXQ)2 /4. If Xo for the initial-value all t. Therefore,

zero

a unique solution through

Xo

at

zero.)))

6

1:

Chapter

Scalar

Autonomous

Equations)

the domain of f(x) == JX is naturally In this example, this situation arises a subset of ffi. In applications, often, cannot population of insects grow to be negative. (:;

restricted

to

instance,

a

for

The above show the necessity of certain conditions on the examples of solufunction f in order to guarantee the existence and the uniqueness We will state such a theorem tions to the initial-value problem Eq. (1.2). in the Appendix. also a more result and below, First, howpresent general we need to a small of notation. introduce ever, piece functions f : ffi ---+ ffi by We will denote the set of all continuous CO (ffi, ffi), and the set of all differentiable functions with continuous first 1 we will use en (ffi, ffi) to indicate derivatives Analogously, by e (ffi, ffi). order n. If the domain the functions with continuous derivatives up through of is a subset U of ffi, then we will use the notation functions CO (U, ffi), etc. If there is no ambiguity, we will usually omit the dependence on the 1 and simply refer to a member of one of these sets as a Co, e , domain continuous function of or en function, etc. In the case of a real-valued k ---+ 1 if all the first several ffi is said to be a e function variables, f : ffi

are continuous.

derivatives

partial

To emphasize xo

== 0

at to

for this

the

1.4.

If f E

infinite)

any Xo E ffi, to !3xo) containing

(Qxo,

Ixo

the initial-value

Also,

for all

==

t E Ixo, satisfying the

if Qxo is finite,

if !3xo is

finite,

interval (possibly '1'(t, xo) of

a solution

==

Xo,) condition

initial

'1'(0,

xo)

==

Xo.

I

==

+00,

1'1' ( t,

x 0) I

== +

00.

(3;o)

1 f E e (ffi, ffi), then 'P(t, xo) is unique on Ixo and derivaits first partial '1'( t, xo) is continuous in (t, xo) together with 1 tives, that is, 'P(t, xo) is a e function. (:;

in addition,

If,

interval Ixo possible largest the maximal interval of existence

The

mal

is an

0 and

then)

lim

called

xo)

Xo.

then)

t-+

(ii)

==

==

Solutions) there

x(O)

f(x),)

lim + I'P( t, xo) t-+a Xo) or,

of

'1'(0, xo)

problem)

x defined

and

x(t)

for

then,

ffi),

==

xo)

and Uniqueness

(Existence

eO(ffi,

often

will

we

'P(t,

words,

x(t) of Eq. (1.2) through use the notation '1'( t,

a solution

of

dependence

initial condition,

solution. In other

Theorem (i)

on the

interval

of existence

of

a solution

in

part of the

(i) of the theorem above solution '1'(t, xo). The maxi-

of Example

1.2 is shown in Figure

is

1.0.)))

1.1. Existence

o)

Maximal

1.0.

Figure

value x(O) = 1 is

initial

and

Uniqueness

t)

1)

interval of existence of the

of x

solution

= x2

with

1).)

(-00,

the function f may not be defined on all of JR. One that f E Cn(U, JR), where U is an open and bounded subset of Theorem 1.4 are the same case, the conclusions \037 \037 t that all of the limit of must as except points c.p(t, xo) Qt o (or t (3;o) to the of U. belong boundary Let us now return briefly to the notation c.p(t, xo) for the solution of an initial-value and reexamine it in light of our foregoing discussions. problem 1 For a given C function f, Theorem 1.4 implies that the family of all specific solutions of x = f(x) can be represented by c.p(t, xo) viewed as a function of two where t E Ixo and Xo E JR. As such, c.p(t, xo) is called the variables, of x = f(x). The domain of this function of two variables could be flow somewhat because the domain of t may depend on Xo, as seen complicated in Example 1.2. The fine structures of flows will be one of our in the main concerns For the moment, we will be content to introduce a chapters. following If f is a C 1 function, common name for our subject-dynamical systems. flow each rise to a the for t, then, map of JR into itself (with gives c.p( t, xo) \037 restricted Here are some of the possibly given by Xo domain) c.p(t, xo). of this important properties map: In applications, is situation of JR. In this

common

(i)

c.p(0,

(ii)

c.p( t

xo) + s,

= Xo,

xo) =

c.p( t,

c.p( s,

xo))

for each

t and s when the

map

either

on

side is (iii)

c.p( t, c.p(

-t,

defined, 1 xo) is a C map

for

each

t and

it has

a C1

inverse

by

given

xo).

itself satisfying these three properties on we can say in conclusion JR. So, dynamical system scalar autonomous differential equation gives rise to a

A map

of

JR

into

is called

that

the

flow

dynamical

a C1 of a

system)))

7)

8

on

are

There

JR.

Autonomous

Scalar

1:

Chapter

will see one such

Equations)

also other

ways

important

case

of

in

and we

systems

dynamical

obtaining 3.)

Chapter

1.1.

.

,. C/

Exercises

Show that the

hypotheses: a unique solution

Verifying o has

the

maximal

x

of

of existence

interval

0, then the solution apor tends to +00 as t \037 (3xo; see Figures 1.5a proaches an equilibrium point if solutions of the initial-value problem and 1.6a. Furthermore, for Eq. (1.1) are unique, then the solutions two different initial conditions with through Thus we have the lemma: < < Xo following Yo satisfy cp(t, xo) cp(t, Yo).

is very

It

In fact, the If f(xo) bit.

lem

(i)

(ii)

is

,+(xo)

tively,

Let

cp ( t,

Qxo

us now We

Yo) for

note,

==

all t if ,-

and

-00]

x is

an

illustrate however,

cp(t,

of the

xo)

initial-value prob-

Then

[respectively,

where

-00],

Xo.

a monotone function

cp(t, xo) is cp ( t, x 0)
0, the only orbit is (-00, +(0), and there is no equilibrium point. We have marked the directions of all these orbits in Figure 2.2. If the c is varied, as long as c < 0, the number and the parameter direction of the orbits remain the same; the is the shifting only change of the location of the equilibrium points :f: yCC . Sirnilarly, for all c > 0, there is only one orbit and its direction is from left to right. However, if of how small an amount c is varied, c == 0, regardless the number of orbits there are two for c and none for c > o. 0) < equilibria 0, changes: any c
1, ticularly Figure accessible diagrams mostof x (mA + 1) x details; in formula is An account of the reference m < 1.) m = 1, andis looss stability given [1979]. were first Wan [1978]. Strong resonances investigated by Arnold [1983] The case Takens and [1974]; an exposition is contained in Whitley [1983]. With these at our disposal, we now turn to generalities.) unresolved. still remains of fourth roots ofexamples unity was initiated by Poincare and studThe study of area-preserving maps Exercises) \037\037.o) for a proof of the ied by Birkhoff extensively [1927]. The main reference 2.1. the of in with orbit Exercise 1.5 the same structure. examples and Siegel is Moser TwistIdentify Theorem groups 1967] [1973]; see also Moser [1962and in a review in Exdescribed diagrams More of recent and Moser [1971]. by developments of thearebifurcation 2.2. Provide the details the computation Hamilto-))) for analytic formulated Moser ample 2.7.))) A comparable theorem was [1986]. usual

or

practice,

it would

B) \"Neimark-Sacker\"

appropriate I .. .. \302\267 . ....

39)

40

2:

Chapter

Bifurcations)

Elementary

2.3. Constant

and

\"logistic

density x(t)

of

such

the

population at a rate

grows

and h

are is

However, when the

- cx 2 -

its size

the

dynamics of the differential equation) The

the

reflect

when the

growth rate

intrinsic

Notice that the

population

density

population

is small.

large, growth is impaired because reflects this behavior.

gets

population

to

according

grows

h,)

c

k and

the

by

of harvesting.

rate to

proportional

== kx

positive;

the

rate.

constant

is governed

a population

all the coefficients

where

at a

is harvested

x

of

a population

that

Suppose

harvesting:

model\"

of,

for

the x 2 term overcrowding; of harvesting k and c, to determine the effect the problem is, for fixed Now, we are the population on the population. Since density cannot be negative, for x > O. For a positive initial of this equation interested in the solutions of value the population is exterminated if there is a finite population density, t such that cp( t, xo) == o. Without finding explicit solutions of the differential example,

equation, show the following: 2 0 < h < k / (4c), then there is threshold ( a) If the harvesting rate h satisfies such that if the initial size value of the initial size of the population On is below the threshold then the population is exterminated. value, then the the other hand, if the initial size is above the threshold value,

population

approaches

(b) If the harvesting rate exterminated regardless 2.4.

an

h

h

of its initial

that a

harvesting: Suppose as in the previous

Proportional

model\"

\"logistic

to the

portional

size

of the

point.

equilibrium satisfies

2

> k / ( 4c),

then the

is

population

size.

grows

population

but

exercise,

to the

according

is harvested

at a

rate

pro-

population:)

x

- cx 2 -

== kx

hx,)

constants. and h are positive Show that if k < h, then, regardless extermination tends toward initial density Xo > 0, such a population as t \037 +00, but is not exterminated in finite time. Also, analyze the fate in the cases k == hand k > h. the population

where

k, c,

of the

2.5.

The

Hydroplane:

rolling,

is

determined

rectilinear by a

motion

scalar mv

==

of a

differential

T(v)

hydroplane,

equation

-

ignoring

of the

pitching

of

and

form)

W(v),)

of the hydroplane, m is its mass, T is the thrust of the is the velocity It is reasonable to assume, for mechanism, and W is the resistance. the thrust is constant. The resistance, on the that approximately simplicity, other should increase with small and large v, but can be negative for hand, of the hydroplane and the intermediate values of the velocity due to rising motions of the hydroplane decrease of the wetted area. Discuss the possible for various values of the constant thrust.)))

where v driving

2.2.

In

this

Theorem

Function

Implicit

Implicit Function Theorem)

The

2.2.

The

a

we state

section,

known as the

turns out

which

Theorem,

mathematical

from

result

fundamental

Function

Implicit

to

analysis be

indis-

an

below version presented pensable tool in bifurcation theory. The simplified of equilibria of scalar differential is tailored for the study of bifurcations A more general form of this important theorem is given in the equations.

Appendix.

Let

-

A

IIAII of A to

(AI, . . . , Ak) be IIAII

we may

which of

vectors

Theorem 0 1 function

a vector in

For

IRk.

2.8.

==

as the

interpret

+ ... +

(AI

will say more

A. We

of

length

F:

that

Suppose

about

norms

IR

x

IRk

---+

(A, x)

IR;

r--t

F(A,

x),

is a

satisfying)

8>

are constants

and)

== 0

0)

and

0

(0, 0) #

ox

II All

< 8}

o.)

a 0 1 function)

0, and

TJ >

{A :

:

'ljJ

---+ IR)

that)

such

'ljJ ( 0)

if there

Moreover,

The Implicit following

== 0

is a

context.

F ( A,

and

(AO,

the equation

satisfies

and

on

\"norm\"

A%)1/2,)

of

there

the

7.

in Chapter

F(O,

Then

we take

now,

be)

xo)

E IRk

F(AO, xo)

'ljJ ( A

))

X IR ==

0,

for

== 0

such

that

then

Xo

II A II
\", x) is guaranteed

existence

Answer:

'lj; ( >..)

==

-

2.3. Earlier

linear

sin x;

with

>..)x

+

>..

-

of

In this section an equilibrium

essentially

be emphasized

of

cos(7r/6 +

the

equilibrium We

the

as in

the

following

(c)

solution

-1).

(1,

sin>..

for

satisfied

+ tanx.)

differential

specific

[Eq.

equations,

and investigated

(2.6)],

of the points as a function differential equation x == if the first term of the that,

a given show

at x is the

perturbations

same in

'lj;(>..)

Equilibria

consider

field f

is a unique

there ==

x);

(2.2)], cubic

point x. vector

the

general

that

we

that (J-L, y)

Function Theorem are

Implicit

Near

bifurcation

and

the function Theorem.

Function

Implicit

2.1 we considered three

Section

Taylor expansion then under rather

x is

(b) 1/2

[Eq. (2.1)], quadratic [Eq.

the stability f(x)

of the

Perturbations

Local

parameters.

the

the Implicit Function Theorem to show equation J-L + (1 - J-L)Y + y3 == 0 near

>.. +

in

by

2

+ x , determine

>...

2.8. Show that the conditions the following functions: (a)

(1 +

== >.. +

examples

analysis

mentioned

we

will

or

quadratic,

linear,

bifurcation

cubic,

of equilibria near above. It should

consider

the

effects

of)))

2.3.

Near

Perturbations

Local

Equilibria

our results are valid Thus, quite arbitrary, but small, perturbations. only a sufficiently small neighborhood around the equilibrium For point. == we shall assume that x has an at 0; if equilibrium point simplicity, I (x) we translate in can to a new coordinate as the not, always proof system, in

of

1.14.

Theorem

I: Hyperbolic

Case with

==

j(O) properties

of the

the higher

order

point

linear

approximation

1

in the Taylor structure of the

perturbations

the

effect

We

=1= o.

j'(O)

that Suppose I is a 0 function saw in Theorem 1.14 that the stability 0 of the differential equation x == j (x) is of the vector field near 0, that is,

equilibria.

equilibrium

by the

determined do not

0 and

qualitative

expansion

field

vector

the

of

zero.

near

flow

the

However,

if we make that influence question remains: what happens perturbations the constant and the linear terms? We will show below that the situation 2.1 also prevails in the general case. in Example To be precise, consider the perturbed differential equation)

x

where F :

x IR \037 IR;

IRk

==

r--t

(A, x)

(2.12))

x),)

F(A,

is a

x),

F(A,

C 1 function

satisfying)

aF

F(O, x)

==

and

j(x)

ax

==

(0, 0)

=1= o.)

j' (0)

(2.13))

Let us first the existence of equilibria of the perturbed equainvestigate tion (2.12). If F(A, 0) =1= 0, then the origin will no longer be an equilibrium point. However, from Eq. (2.13) and the fact that j(O) == 0, we have) == 0

F(O, 0) the

Hence,

8 > 0 and

Implicit 0, and

TJ >

aF

and

==

(0, 0)

ax

Function Theorem a 0 1 function 'ljJ(A)

I' (0)

=1= o.)

that

there

implies

for

defined

II All

constants

are

with

< 8

== 0

'ljJ(0)

such that)

F(A, every

Moreover, given

by

(A,

(A,

The

'ljJ(A)).

stability

Theorem

with

x ==

'ljJ(A)

behavior 1.14.

8 and


0 such

Thus,

'ljJ(0)

that,

(2.14))))

'ljJ(A)).

(A,

for

==

0, we

have

II All < 8,

\037; (0,

'ljJ(0))

==

the sign of Eq.

1'(0)

=1=

(2.14) is)

43)

44

2:

Chapter

Bifurcations)

Elementary

the stability type of the equilibrium the same as that of 1'(0). Therefore, is the same as the stability type of 'ljJ(A) of the perturbed equation (2.12) the equilibrium 0 of the unperturbed equation x = I(x). We can summarize the discussion above by saying that the flow near

to

a hyperbolic equilibrium point is insensitive vector field.

small

of the

perturbations

Case II: Equilibria with quadratic that I Suppose degeneracy. with 1(0) = 0, I' (0) = 0, but I\" (0) =1= o. This is the next order of complication that occurs when we cannot a decision about make the stability of an equilibrium on the based linearization. point Let us consider the perturbed differential equation) a

is

0 2 function

x = F : IRk x

where

F(O, x)

sion

IR

---+

= I(x),)

These conditions of F about the

8F

8

>

II All

(0,

8x

0,)

2F

( 0, 0 ) 8x 2 =

a(A)

and

its

=1= o.)

(2.16))

that the Taylor

expan-

2

+

c(A)2

for any

0,

and,

G

satisfies

the

x),

G(A,

IG(A,

t > 0, there x)1 < tlxl

differential

(2.2) depending on one

perturbation

satisfying)

form:)

a(O) = 0, b(O) = 0, c(O) = I\" (0) =1= 0 and 'TJ > 0 such that the function < 8 and Ixl < 'TJ. As an instance of Case II, let us recall

2 I(x) = x

function

= I \" ( 0 )

0 imply

x b(A)X +

+

02

is a

x), 8

=

0)

(2.15))

x),)

F(A,

together with f(O) origin has the following

F(A, x) = with

\037

(A, x)

IR;

F(A,

parameter

for

x =

equation (k

are

2

=

1):)

2

X =

F('x, x)

in Example 2.2 that two equilibria when Af\"

the

=,x + f\"(O)

.) \037

of this equation changes at A = 0 to no equilibrium when Af\" (0) > 0; behavior of this see Figure 2.2. We will show below that the bifurcation for the general case [Eqs. (2.15) and (2.16)]as well. simple example occurs To verify the assertion above, it is only necessary to demonstrate that = 0 for any small A has a graph which is like a the function F( A, x) near x the existence of a unique parabola. This can be accomplished by showing extreme point of the function F ( A, x) near x = 0 for small A. The extreme F of to the solutions x of the points correspond equation) We

from

saw

(0)

flow

< 0

8F

8x

(A, x)

=

O.)

(2.1

7))))

6)

On

and

Tori

generalization of the ideas from Secif a I-periodic nonautonomous differential is also periodic in x, then it gives rise to a equation on a torus (the surface of a doughnut). differential equation The dynamics of such equations are explored most convein terms of their Poincare which to maps, happen niently in the spirit of Chapter 3, we include a be maps on a circle. Accordingly, brief discussion of such maps and study a landmark the standard example, circle map. Poincare, in conjunction with his work on classical mechanics, was of differential the first to study vigorously the subject the flows for these equations two cases on a in circle his a Since particular maps. days, 2.2). deep These analyticalresults theory several values of a(A) (compare with can of with torus, Figure circle has The of this is to maps analytically emerged. purpose that whenchapter are out be summarized < 0 therepoint by saying a(A)/\" (0)merely a few facts and some will return to this We = rudimentaryequilibria near the origin,highlights. two hyperbolic is a 0 implies that there subject a(A) in Part IV and oscilfrom the > theory seminalandexamples explore when a(A)/\"(O) atseveral the origin, 0 there ofare equilibrium nonhyperbolic lations and Hamiltonian where tori are mechanics, omnipresent.))) about the naturally no equilibrium points origin. in the discussion It is important to observe above that the qualitative of the flow of the perturbed is determined from a structure equation (2.15) of the the function function parameter A, namely, single a(A) correspondof F(A, x). Thus, even there value though may be k ing to the extreme = . . . the bifurcation A of the , Ak), parameter components (AI, A2, (vector) of the perturbed on a single number, behavior equation (2.15) depends we occurs in a the When this situation bifurcation problem, say that a(A). original vector field 1 is a codimension-one bifurcation.))) In

this

chapter,

tion 4.3, we

show

as a

Circles)

that

46

2:

Chapter

Bifurcations)

Elementary

2.9.

Example

Two

consider the

differential

of a vector

A

example

x2

A2X +

A2) are two to the corresponds it is and , given by) ==

(AI,

==

the

Thus

two

+ A2 X

Al

small

parameters,

2 + x ,)

==

Al

points

of

for this

a(A)

function

the

x)

F(A,

==

Al +

\037A\037.)

curve

the

cross

we

equilibrium

function

The

parameters.

A2)

occur as

bifurcations are

a

of

example

on two

depending

minimum value

a(AI'

plane: there

field

equation)

x

where

one: As an

codimension

but

parameters

bifurcation

codimension-one

if

Al


none

and

A\037/4,

0)

A\037/4.

In applications, a single affect several parameter may of a the For expansion instance, consider perturbation. Taylor == x 2 of the vector field perturbation given by) f(x) 2.10.

Example

terms in the following

2 x == A

A is

where bifurcation

a scalar

parameter, and a of the

behavior

+ x2 ,)

+ 2aAx is

a

As before, the

constant.

given

equation depends 2 + 2aAx + x , which

perturbed

minimum

the

on

== value of the function F(A, x) == A 2 is equal to a(A) 2 2 term in the the linear A (1 a ). The sign of a(A) is determined by perturbation if lal > 1, and by the constant term if lal < 1. The bifurcation takes at A == 0 in both cases, but the flow is different depending on place whether lal < 1 or lal > 1. You are invited to draw the bifurcation diagram of the flow. 0 and representative pictures

Case III: Equilibria with

03-function

f (0)

==

0,

the level of complication

that cannot

3 that -x

==

f' (0)

equilibrium point at zero order terms of the Taylor We

cubic

with

occurs

degeneracy. I\" (0) == 0, but if the properties

expansion of f(x). (Section 2.1) in

have seen previously

at least two

were

parameters

f'\" (0)

of the from the linear

be determined

the

In particular,

=1= O.

or

all

dx -

we

of

the have

a

is

is

This near

flow

example

specific

to capture

needed

types of behavior of I under perturbations. in the perturbed the bifurcations equation

that I

Suppose

0,

the

second-

f(x)

==

possible studied

x 3 ; see ExamWe will show below that this example ple 2.6 and Figures 2.12 and is representative of what can happen in the general case near the equilibrium point x == 0 of I satisfying the conditions above when I is subjected are small together with their derivatives to \"nice\" perturbations that up 2.14.

through

order

three.)))

x

==

c +

2.3. More

the

consider

specifically,

perturbed

x= F :

where

x IR \037 IR;

IRk

and would plicated to the two-parameter

important

analysis

of

C3

function

case. In the

= fill (0)

satisfying)

and

perturbation,

to this

lengthy

com-

rather

ourselves

perturbation.

special

section

technical

is

confine

we first study an then show how to

below,

perturbation and

we

Therefore,

presentation

two-parameter

of:- O.)

and (2.19)

(2.18)

Eqs.

far afield.

too

two-parameter

three-parameter perturbation. the bifurcation present analysis perturbation F(A, x) with the following

by giving

a simple

A2),

a

for

A = about

two-parameter, expansion

Taylor

origin:)

F(A, x) with

c(O) 0 such

'TJ > Ixl

0)

(0,

first

We

(AI,

is a

of a

example

the

equation)

(0, 0) = 0,

ox

\037;:

this somewhat

We conclude

of

0,)

us

take

particular

a general

reduce

=

2 (0, 0)

bifurcation

A complete

differential

47)

Equilibria

(2.19))

o2F ox

Near

(2.18))

x),

F(A,

= f(x),

F(O, x)

Perturbations

x),)

F(A,

r--t

(A, x)

Local


O.

(0)

(2.20) is an

Equation of

'l/Jl(X) =

by

(2.21) and (2.22) near the

of a cusp in the (AI, are the parametric representation equations coincides with the curve) which near the origin approximately

These plane

f'\"

of Eqs.

49)

Equilibria

by)

A2(X) =

Eq.

computations above

A2(X)

and

Al(X)

If

of the

results

the

Near

Perturbations

Local

of a

example

field with cubic

a vector

perturbation

\"good\" two-parameter

degeneracy in the

that

sense

the

bifurcations

and the linear terms of the Taylor are determined only by the constant The term of the vector field. expansion C(A)x2/2 did not enter into the in the see Eqs. (2.25) and first to the cusp approximation (AI, A2)-plane; in A and c(O) = O. function is This is because the differentiable C(A) (2.26). the Taylor

Thus,

expansion

for

= ClAl

C(A)

Cl and

where

are

C2

hence,

constants;

C(A)X

2 =

be given

must

C(A)

+

C2 A 2

the term +

ClAlX2

+ ...

by)

,)

has

C(A)x2

C2A2X2 + . . .

the

form)

.)

is smaller than AI, and the x is small, the term ClAlX2 it is to be expected that the than A2. Therefore, bifurcations of Eq. (2.20). term C(A)x2 has little influence on the We now show that an arbitrary two-parameter perturbation of the can be reduced to cubic under certain reasonable conditions, degeneracy, Let J.l = (J.ll, J.l2) be two the special two-parameter perturbation (2.20). Observe

term

when

that,

C2A2X2

is smaller

parameters and

consider

about

the

expansion

F(J.l,

with are

b

x)

a(O) = b(O) > 0 and 17

11J.l11 < band

Ixi

=

the

X2 a(J.l)

=

with the following

F(J.l, x)

perturbation

Taylor

origin:)

+

2(0)

>

0 such


1. This suggests the following

lal

point.

3.8.

Theorem

stable

any

xl < 8, the

from

fixed

the

fixed

Proof.

the

of the

type

evident

unstable

linearization about

given in Theorem 1.14, we expect, of the fixed point x of a type

stability

stability

that a

with the

equation

is

stability

be

analogy

that the

if, for

stable

be

which

Definition 3.7. A fixed point x of J is said to it is stable and, in addition, there is an r > 0 n \037 +00 for all Xo satisfying Ixo - xl < T.

the

of

we make

equations,

said to

for

the inequality IJn(xo) - xl < if it is not stable. unstable

Xo satisfy

x is said

It

notions

to the

definitions:

following

of

of equilibria

to fixed points and in-

return

Analogous

properties.

stability

us

let

diversion,

if

IJ'(x)1

For

Let J be a C I map. A fixed point < 1, and it is unstable if IJ'(x)1 >

convenience,

the origin (0, 0).

Let

u

we first translate the point the new variable defined

be

x

of

J is

1. (x, by

x) u

asymptotically ==

(x,

- x-x.

J (x))

to

Then)))

73)

74

Chapter

3:

Scalar

Maps)

Xn + 1) X2)

Xn)

45\302\260 Line___)

a = 2.0)

Xo)

a = 0.5)

X1)

Xo)

a = -0.5)

X2)

a = -2.0

Figure

3.3.

Typical stair-step diagrams of linear map

Xn+l

== aXn.)))

3.2. the map

f

f(x + u)

g(u) Clearly,

g(O)

to

equivalent

f(x).)

studying the stability of the stability of the fixed (x + u). 0 and define)

E >

fix

min

m\037

+ s)l,

If' (x

max

M\037

Isl::;\037

Since

=

g(u)

if


1,

Suppose

that

Since

EO.

point

80)nl u l,

shows that there must be a u can be taken arbitrarily zero of 9 is unstable.

Let us return

to

J2

computing

with

difference equation (3.5) is

of the

value close

New-

equiv-

map)

2

3.4 the

+

1

x

'

of

graph

this

orbits, one with

that there

two

approximate

o.)

m\037o >

(1 +

inequality

solutions

in Figure

course,

n >

---+ +00.

of its positive from

of

that

>

x

diagram

3.6, that

0 such

f(x)= We

1. Furthermore,

M\037
zero, this implies that the fixed

alent

that

have)

we

asymptotically stable. part of the theorem observe that,

of n, to

>

n

for

0 such

Definition

E in

M\037
as

M:lul) E >

1, M: so the fixed point is

prove

there

0 and

i=

8=

since

Also,

and

---+ +00

M\037lul.)

< M:E
..)

small.

To

define

T(>\",

x), we

use the

F(>\",

T(>\",

Here

is a that)

a(>..)

+

+ b(>\x")

x):)

x).)

G(>\",

that of the theorem imply small. b( >..) i= 0 for >.. sufficiently = 0 is equivalent to the equation x = T(>\", x) x) with)

The hypotheses 1'herefore, F(>\",

pose

x) =

of F(>\",

expansion

Taylor

x)

F(>\",

x) =

-

x).)

b(>..)-lC(>\",

the convergence

illustrating

example

specific

= -b(>..)-la(>..)

>.. +

(1 +

+ x

>..)x

of

the

iterates.

Sup-

2 ,)

>.. is a scalar Then the function parameter. 'ljJ(>..) of the Implicit Function Theorem is 'ljJ(>..) = ->... Recover this function the method of using For this purpose, that successive approximations described above. compute = ->\"(1 + >..)-1 - (1 + >..)-lX 2 . Then, take>.. = 0.1, >.. = 0.3, >.. = 0.5, T(>\", x) in initial value Xo = O. Do you observe any difference etc., and iterate with the rate of convergence for different values of >..?)

where

A

aspects

study of this 5. Chapter

We {XO,

of maps,

class

restricted

certain

be

of Monotone

Bifurcations

3.3.

begin

of

differential We

class.

will

some

introducing

by

of f,

Xl, X2, ..., ...} monotone nondecreasing X n ,

Maps)

called monotone maps, play In this section equations. apply the results to differential

if

where

terminology. Xn+l

==

f(xn)

the

a central we

undertake

role in the

equations

in

A positive orbit ,+ == for n > 0, is said to is nondecreasing, that

{xn} > X n for every positive is, Similarly, ,+ (xo) is said to be Xn+l < if monotone X for n every positive integer n. Combinnonincreasing Xn+l these two we that notions, simply say ing ,+ (xo) is monotone if it is either

monotone nondecreasing monotone

map

if every

or

sequence n. integer

monotone

nonincreasing.

positive orbit ,+ (xo)

of

f

is a

Finally, we call f monotone sequence.)))

a

81)

82

Scalar

3:

Chapter

Maps)

If f is a C 1 function with f'(x) 3.14. > 0 for all x in the domain of definition of f, then f is a monotone map, that is, the positive orbit of is a initial condition monotone Xo sequence. ,+ (xo) any From the Mean Value we have Theorem, Proof. Lemma

-

Xn+l

for some

xn.

For the

-

Xn+l

of

purposes

for

X n ,

f to

we will require

dynamics,

X n

-l))

n,

positive integer

every

- Xo. 0

Xl

-

= f'(xn)(X n

n -l)

f(X

C

least

at

be

the

has

1

with

and refer to such an f simply as a monotone map. is monotone, then the inverse of f, exists. We will use the f-l, the n-fold of f-l with itself. composition f-n to denote

derivative,

positive

If f notation

If f is

3.15.

Definition

of points

of Xo

of

-

f(x n )

Therefore,

that

sign as

same

=

X n

f-l

Xo,

defined

is

monotone, then the negative . . ., and is denoted

(xo), f-2(xO), to be ,(xo)

,+

is

Xo

the

set

The orbit,

(xo).

(xo).

U,-

(xo)

of

orbit by,-

The geometry of orbits of a monotone map is very similar to that of a scalar differential the fixed points act like equilibria, and we can equation: use arrows to indicate the direction of other orbits under forward iteration. to study bifurcations of fixed of monotone maps we points Consequently, need the results in Section as we shall do now. 2.3, only to reinterpret For of let us assume that the notation, map simplicity f has a fixed = if at x we can coordinates to make it so. Fur0; point not, change so in that 0 that is monotone a > thermore, suppose f sufficiently f'(O) small neighborhood of the origin. the perturbed map F(A, Consider x) on k parameters

depending

F : If

in x

the

is that of

of

analysis

of the

zeros

the

\037

(A, x)

of

value

small

fixed

A2, . . . , Ak):

(AI,

C 1 function, then

is a

each

for

x JR \037 JR;

JRk

x)

F(A,

A

A.

it

that

follows

each

for

Now,

of F(

points

with

x)

F(A,

A,

x)

F(O, x)

F(A,

fixed

x)

A,

the

is equivalent

is

= f(x).

also

key

monotone

observation

to the

analysis

function)

x)

F(A,

have

-

x.)

the bifurcations of zeros of a function, of under various types of hyequivalently, equilibria, on the linear, quadratic, and cubic terms. We now translate those potheses results for bifurcations of fixed points of monotone maps.) In

2.3 we

Section

the

or,

Case

C1

map

I: with

analyzed

bifurcations

f(O) = 0 and

F ( 0, x )

Points.

Fixed

Hyperbolic

f'(O)

= f (x

=11.

aF )

Suppose

Consider

and

ax

that

a C 1 map

(0, 0 )

= f , (0 )

f is

a monotone,

F(A, x) satisfying i= 1.)))

3.3.

Figure

3.8.

stable

Asymptotically

O.5x persists

A +

as

A is

hyperbolic

of Monotone

Bifurcations

fixed point of

F(A, x)

varied.)

the perturbed Then, for IIAII sufficiently small, map F has point near zero whose stability type is the same as the stability point zero of the unperturbed map f.) 3.16.

Example

map

f(x) =

and

its

one-parameter

F(A,

of the parameter For each value whose stability type is the same for

the

the

diagrams

stair-step

parameter

linear map: Let us consider perturbation given

One-parameter

O.5x

A.

0)))

Maps

x)

of the

the

fixed fixed

linear

by)

= A+O.5x.)

A, there is a unique the fixed point

as

of the

a unique

perturbed

map

for

F

fixed point hyperbolic A = O. See Figure 3.8 for several

values

of

83)

84

Scalar

3:

Chapter

Maps)

2 monotone, C

f is a a Consider

C2

there

Then

extreme

the

The

( 0, 8x

82 F

0) =

1,

\"

f

points

one

fixed

point

of F,

no

fixed

point

of F

(0)

values

3.17.

===>

> 0

of

= 0,

0:(0)

===>

which

A.

map f(x) = it is monotone.

the

case,

of

f

given

by

2

F(A,X)=A+X+X

3.9. In this

in Figure

to

corresponds

of F,

quadratic map: Consider small neighborhood of zero so that of the one-parameter perturbation

bifurcations

=1= O.

that

One-parameter

a sufficiently

Suppose

f\" (0)

( 0 ) =1= O.

fixed

illustrated

are

0) =

0, 8x 2 (

two

===>

= 0 ( A) f\" (0)

small

local

1, but

(0) < 0

sufficiently

in

=

(0)

x) - x such

O:(A)f\"

x + x2

= 0, f'

f(O)

is a function O:(A), satisfying value of the function F(A,

0:

Example

8F

= f (x,)

O:(A)f\"

for

Degeneracy.

Quadratic

with

map

x) satisfying

F(A,

map

F ( 0, x )

with

Points

Fixed

II:

Case that

we

have

O:(A) =

A and

Therefore, when A < 0, there are two fixed points; at A = fixed point; for A > 0, there is no fixed point of F(A, x). 0

f\"(O) 0, there is

= 2.

one

with Cubic Degeneracy. Suppose that f f' (0) = 1, f\" (0) = 0, but f'\" (0) =1= o. A complete bifurcation analysis of an arbitrary C 3 perturbation F(A, x) of of Therefore, we confine our discussion to the analysis f is difficult. a \"typical\" of a two-parameter example perturbation. As in Case III of Section 2.3, under mild of fixed hypotheses, study of local bifurcations F of can reduced to this The be points example. map any two-parameter details are identical to the ones given previously. Points

Fixed

III:

Case

is a

3 monotone, C

map

with

f(O) = 0,

3.18. Two-parameter cubic Example map: - x 3 in a small of the neighborhood origin the two-parameter, A = (AI, A2), perturbation

Consider so that f

x

F(A, x) = there is a

. . . The trated

the

cusp

in

are

three

there

Al

+

(1 + A2)X

(AI, A2)-plane such that fixed points of F(A, x) for

the map is

f(x)

monotone.

==

For

of f given by - x3 of (AI, A2)

values

inside the

cusp, there

is

one

fixed

point of F(A,

x)

for

values

of

(AI, A2) outside

the

cusp, are

there

if (AI, possible

two

fixed points of F( A, x) for values of (AI, A2) on the cusp = (0, 0). 0), and one fixed point if (AI, A2) bifurcations of fixed points of F listed above are illus-

=1= (0,

A2)

local

in Figure

3.10.

0)))

3.3.

Figure 3.9. Bifurcations origin: values of A are

of fixed

-0.1,

points

0, and

0.1.)))

Bifurcations

of F(A, x)

== A

of Monotone

+ x

2 + x

near

Maps

the

85)

86

3:

Chapter

Scalar

Maps)

A1)

Figure

3.10.

of

Bifurcations

F()\",

x)

=

(1 +

)..1 +

)..2)X

- x 3 .)

if f is not monotone, its hyperbolic Even fixed points persist under small perturbation. Nonhyperbolic fixed of points f, however, can undergo no counterparts bifurcations with in our catalog of bifurcations of equilibria of scalar differential now We turn to one such of bifurcation equations.

importance.)

great

Exercises)

3.12.

4CV.O)

A trans critical bifurcation: a transcritical undergoes

this map with the 3.13.

A saddle-node a saddle-node

differential

Show

that

the map

bifurcation at the equation

bifurcation: Show bifurcation at the

that parameter

parameter

in Example the

map

F()\",

value)..

x)

F()\",

value)..

=

(1 +

+ x )..)x = O. Compare

2.3.)

x)

= eX -

).. undergoes

= 1.

3.14. Find a value of the parameter).. at which the map F()\", x) = ).._x 2 undergoes a local bifurcation. Identify the bifurcation and draw three representative to illustrate your bifurcation.))) stair-step diagrams

2

3.4.

3.4.

Bifurcation

Period-doubling

In this bolic

we

section,

fixed

to perturbations. As

we

bifurcation that a nonhyperwhen f is subjected

an important

investigate

x with

point

87)

Bifurcation

Period-doubling

f'(x) = -1

is

to undergo

likely

saw in previous examples, it flips a point close to

monotone and

when f'(x)
0, there period 2. Furtherorbit is asymptotically stable [respectively, unstable] if more, the period-2 at the origin is an unstable [respectively, asymptotically stable] fixed point this value of A. Proof. fixed

Periodic points

F 2 (A, x) a zero of

points of of F 2 (A, x)

period 2 ==

of

the

A, F( A,

F(

map

F(A,

x) correspond

x)), equivalently, to

the

to

zeros

the

of

- x. However, because of condition x == 0 is point (iii), the fixed this equation but its minimal period is 1. Therefore, to avoid this and locate only the periodic of minimal period 2, we need to points

point analyze

zeros

the

of the

function)

1

-

x)

2

[F

(A,

x)

-

x] .

To the first several accomplish this, we begin, as usual, by determining terms of its Taylor expansion. Let us use the notation to denote \"prime\" the derivative with respect to x and compute some derivatives:)

2 [F

the 2two

(A, x)]'

==

F'(A,

F(A,

x)) F'(A,

x),)

For any[F'(A, is a unique intervals. Xo in+ (-00, point x)]2 Xl), there F\"(A, F(A, x)) F'(A, F(A, x)) F\"(A, x).) = on such that t 01. If we let Xo depending cp( XQ ' xo) In particular, then h we is ahave) homeomorphism. To extend h, let h(Xl) = (3l), origin, h(xo) = 1/J( -t XQat' the r--t Xl r--t as Since now the map h : (-00, Xl] ---+ (-00, Xl. Xo Xl, h(xo) Xl] 2 2 2 == == == F 0 ==0.))) so defined is a 0, 1,) ( ( ) homeomorphism.))) 0)]' [ [f ]' [F (0, 0)]\" [f2(0)]\" of

value

[F

open (A, x)]\"

t XQ

of

==

time

3.4. It follows from these is given by) origin

the

that

formulae

of F2(A,

expansion

Taylor

89)

Bifurcation

Period-doubling

x)

about

the

2

F

a(A) and

functions

the

where

= (1 +

(A, x)

=

a(O) the

Therefore,

1

2

x [F Since

the of

are

the zeros,

If the

x

0,)

=

b(O)

+ . . . ,)

we have

.)

[f2 (0)]'\" been

a(A)

x

is)

seeking

b(A)

x

+6

2

+....)

(3.14))

analysis

slope

A

in a

(1 +

< 0, to determine the cubic function

(0)]'\"

[f2

consider

we

neighborhood A)2

of

of

the

origin,

Example 3.22.

Continuation

of

3.12 and

in Example

its

and x\037,

Similarly,

its

(1 +

if

0 is

A)2 >

1,

(;

Let us now return

to

the

given

perturbation

one-parameter

x\037).

be greater

is stable.

orbit

three

F(A, 0 and


0 and 0 is unstable, then the period-2 map

3

b\037A)

x\037) }.

case

orbit,

simple

the

F(A,

{x\037,

period-2

+

of the zeros of this function is identical to in II of Case Section 2.3 for bifurcations already given In fact, equilibrium points with quadratic degeneracy. > 0, then there is no zero of Eq. (3.14). If A [f2(0)]'\" < 0, to a single period-2 two zeros of Eq. (3.14) which correspond

nonhyperbolic

In

2

satisfy)

b(A)

(A,X)-X] =A(2+A)+T

if A [f2(0)]'\"

orbit,

x

have

we

then there

a

X +

\037A)

expansion

Taylor

=1= 0,

b( 0)

one

2 A)

by

= -x

f(x)

Since

f2(x)

easy to

= x -

verify

satisfied. Thus, orbit

of

minimal

and 3.12. We

1.

For

- 3x 2 ,

18x3 all

that

each

for

period

x) =

F(A,

-(1 +

-

A)X

27x 4 , we

-

of the small

have [f2(0)]'\" remaining conditions

positive

2 which is

value of

A,

= of

A)X

-108

asymptotically stable;

.

=1= O.

3.21

Theorem

is a

there

2

(3 +

It

is are

unique periodic see

3.11

Figures

(;)

conclude

this section

notational

with

simplicity,

several

remarks

we assumed

x =

regarding this

0;

3.21. always be

Theorem can

achieved by translation. 2.

The

nonvanishing

assumption

placed by a condition is always satisfied.)))

on the

on the third second

derivative

derivative

because

be

cannot

[/

2

(0)]\"

re= 0

258

Linear

8:

Chapter

Systems)

.... . \" ,) \037....,.......-)

....................... .................... ............ ..) t)

\\y>) /\302\273){).,

!::,i!\"!!:::f::!:':!! .\".\" \". ...................) ...................... . ....................... ....................... ...............)

.:.:::.:.:.::::::::::::.:....... \" ,) \0370\037jliillli)

Poincare map

8.11.

Figure

The behavior of of the iterates of the

of Eq.

solutions

C : ill?

(8.32),

see

you

linear will

Poincare

undertake a detailed

that

its stability

generalized to

are easily

the

dynamics

map) xO

\037 C

of the

map

of planar

study

ones, in Chapter 15. For find the few remarks below

and

point

xO

ill?;)

in

reflected

,)

linear system

I-periodic

8.11.

Figure

We will the

as the

be viewed

should

which

\037

(8.32) are linear

two-dimensional

linear system.)

a l-periodic

of

have

we planar

the

moment,

acceptable.

developed This maps.

maps, including, of course, however, we trust that The notions of a fixed

3 for

in Chapter essentially

scalar

maps

entails replacing in several norms, Definitions 4.9 and

scalar with and absolute values with quantities vectors, definitions in Chapter 3. With similar replacements, of stability, 4.10 are also readily generalized to yield definitions asymptotic of the I-periodic system (8.32). stability, and instability of a solution A periodic to a fixed point of the solution of Eq. (8.32) corresponds linear and the of the planar map periodic solution of stability type (8.34), is of the fixed the same as the point Eq. (8.32) stability type corresponding in particular of the map (8.34). Notice that the zero solution of Eq. (8.32) to the fixed the of C at the origin. We now summarize point corresponds main implications of these remarks for the zero solution of Eq. (8.32).

Lemma in Eq. (i)

If

Let Then (8.33). 8.22.

IJ.-lil

1, for i




== 2

-..\\.)

(3.16))

that the following lemma shows occurs for initial conditions in

map

f(..\\,

x).)

f'(..\\,

..\\,)

the

4,

logistic

that

Suppose

==

0)

(3.15): > 1, then

0 or Xo


O. Thus, The case 0.5 < Xo < 1 follows from the same argument by noting

We can say limn\037+oo

manner

this,

first

if

Xo

n

\037 +00.

=I-

==

the first

that

iterate

the

about

more

f(..\\,

that

xo)

lies

in the

interval (0, 0.5).

Consequently,

it

one iteration, at most 2, that after (3.16) see Figure 3.13. solutions approach x). monotonically; to it If 2 < A < 3, then x). is still attracting globally but the approach from see Figure 3.14. is no longer monotonic (this is expected linearization); in this case is somewhat more difficult. You should determine 'The proof in the intermediate case ..\\ == 2. what happens is evident

=>

..\\ == be

from Eq.

3 : The

determined

when

1
1, then the solution is again defined for all t. Otherwise, 4.2. For to == of0, X, then the solution is defined only on a finite interval; see Figure we are led to the following of two with invariant a periodsets:))) of 27r. I)))) that solution with Ixo definition notice every I < 1 is periodic

110

4:

Chapter

Scalar

Nonautonomous

Figure 4.2.

Figure

4.4.

Example

Periodic

4.3.

Equations)

flow of

limit: Consider

this equation is given

t.p(t, to,

Notice approach

xo) =

e-(t-to)[xo -

-x +

the

4.5. Limit is

no

-x + cos

differential

t.)

equation)

t.)

cos

by +

\037(sinto

costo)]

that all solutions have the same asymptotic the periodic solution \037(sin t + cos t);

Example

= (cos t)x 2 .)

of x =

Trajectories

x = The

of x

Trajectories

.

x = -x

+

1 1 - - t

fate see

+

\037(sint

Figure

for t 4.3.

the differential

Consider

solution:

+

t 2)))

cost).

---+ +00

: they

I)

equation)

4.1.

Figure 4.4. for

x E III

and t >

1. The r.p ( t,

solution

( t-t 0 )

Xo

that

a solution

solution approaches

every

differential

the

of

as

zero

t

---+

see Figure

equation;

+

to )

( Notice

+ lit - 11t2 .)

1

-

of Solutions

Properties

xo) is given

t.p(t, to,

= e_

to, xo)

= -x

of x

Trajectories

General

by)

1 t,\"

but

+00,

x(t)

== 0

is not

4.4. I)

little that can be said about the general qualitative propdifferential of arbitrary nonautonomous scalar equations. in t, we shall see shortly that However, when the function f(t, x) is periodic case is very similar to the autonomous the qualitative behavior of solutions as is evident in if we replace equilibrium points with solutions, periodic we will derive of this 4.4. Before delving into the details subject, Example nonautonomous linear formula for solutions of a general a useful explicit is very

There

of solutions

erties

differential

equation.

4.6.

Example

Variation

x) nonautonomous

x where a(t) of Eq.

and

(4.2)

is

tial

when

function

constant is

b(t)

formula is ==

0, the

multiplied varied

Consider the

formula.

linear

(in

+ b(t),)

a(t)x

(4.2))

functions.

The solution t.p(t,

to,xo)

by

ft du = e Jj '0 a(u) xo)

This imposing because

==

continuous

scalar

are

b(t)

given

to,

r.p(t,

constants

the

of

equation)

with

t + [xo the

called

e

i to)

constant.

a function

of

t.)))

du

b(s) dS] .

( 4.3))

of the constants formula solution is an exponenthe case,

variation

homogeneous

by a

fS Jj '0 a(u)

In

the

nonhomogeneous

case,

the

111)

112

Nonautonomous

Scalar

4:

Chapter

Despite its defined

formula

looks,

we

(4.2),

ft

If x(to)

==

then

Xo,

y.)

equation (4.2) in the

the differential

Yo and

==

y(to)

a(u)du

== eJto

-

.

== e

y

t

to

respect

of this

solution

the

obtain

with

back to

Now,

returning

enjoy

performing

some

of

the

ft du J to a( u)

differential

b( t ) .) we

equation,

integrate

simply

:)

==

y (t)

t - fS a(u) du e Jjto

Yo +

the

i

variable

we recover

x,

you may

integrations,

we have

solutions

explicit

ds.

b( s )

to)

the

in

used

the solution (4.3). If you this formula to obtain above. 0) examples

use

to

wish

,.(/.0)

Exercises)

4.1. In Example 4.3, x

the

4.2. Write

of

variation

x(O)

x( t)

4.5.

and only Another

l if

+ b(t),

(a) If

Jo

(b)

If

maximal interval

discuss the

and

of xo.)

for the equation x

formula

==

b(t),

where

ft a(u)du = e Jj '0 xo

of

solutions

the

==

a(t)x,

t

e

I8

of x

==

au()

with

f(x( s))

-(sin

there

==

+ f(x)

a(t)x

t)x + 1

ds.)

as t --t +00. solution

a nontrivial

is

0 < t

of x

du

I0

Show that x

problem

a(u) du

t +

solution

the

that

Show

< 1, satisfying

x(O)

==

of the x(l)

l Jo a( u) du

==

if

== O.

the boundary-value Consider problem x boundary-value problem: == Prove the following: with 0 < t < 1, satisfying x(l). x(O) a( u) du i= 0, then there is a unique solution.

a(t)x

l

Jo

== 0

to

function

constants

problem:

boundary-value

boundary-value

4.6.

as a

function.

the behavior

Discuss A

t )x , take

(cos

for linear equations: == Xo satisfies)

Formula

with

==

continuous

is a

b( t)

2

of solutions

of existence

4.4.

new

becomes)

variable

4.3.

To

derive.

new variable y

the

introduce

by)

X

To

fact rather easy to

is in

(4.3)

in Eq.

term a(t)x

the

eliminate

Equations)

0, then

there is a 1

eJ.'

1

solution

a(u) du

b(s) ds

if and

=

O.)))

only

if)

==

4.2. How

exist in this

solutions

many

boundary-value problems: Discuss in the previous exercise for = = et l ; 1, (a) a(t) b(t) = sin t, a(t) b(t) = sin t; (b) = sin 27ft, (c) a(t) b(t) = sin 27ft;

problem

(d)

4.2. In

= sin

a(t)

we begin our

section,

tonomous differential we

study

x= Briefly, assume

Let us

1;

class of nonau-

special

important

the function t with a period

with

is

an

f(t, x) of

the

has

More

1.

addi-

specifically,

equation)

x))

we say f(t, x) the period to be

of

in

differential

f(t,

following

of the boundary-value specific coefficients:

where

equations

the

consider

will

solutions

Equations)

property that it is periodic

tional

the the

t.)

of Periodic

Geometry this

= sin

b(t)

27ft,

Equations

case?)

More

4.7.

of Periodic

Geometry

l-periodic if not,

+ 1,

f(t

x) =

f(t,

(4.4))

x).)

in t. For simplicity we can always rescale

of

we

notation,

t to

make it so.

with certain will be general observations; specific examples in the subsequent sections. of f, the solutions Because of the periodicity of Eq. (4.4) possess certain properties which are useful in determining the asymptotic behavior of the solutions. In order to take of these properties listed below, advantage we will assume throughout this chapter that of Eq. (4.4) are the solutions defined for all t E IR. It is easy to see by direct substitution that if x( t) is a solution of Eq. (4.4), then for any integer k, x(t + k) is also a solution. Let t.p(t, to, xo) be the solution of Eq. (4.4) through Xo at to. The observaof initial-value tion above, in conjunction with the uniqueness of solutions that) problems, implies proceed

forthcoming

t.p(t

t.p(t + The

geometric

translated val

1, to,

+

xo)) on the

2]

1, to

+ 1, xo)

1, to, xo) =

interpretation

horizontally,

[to + 1, to

+

cp(t,

to,

=

t.p(t,

to,

cp(to +

xo),)

1, to,

(4.5))

of Eq. (4.6) is illustrated in Figure

the piece of the solution t.p( t, to, xo) on with coincides the piece of the solution t.p(t,

interval

[to, to

+

1].)))

(4.6))

xo)).)

4.5: when the interto,

t.p(to +

113)

114

Nonautonomous

Scalar

4:

Chapter

Equations)

x)

to ,

( t, \037

+ 1 , to,

\037 (to

x0

))

/ + 1,

(to , \037 (to

C/J1)

to,

Xo )))

(to,

Xo ))

/ iP(t, to,

In the

invariance of

Translation

4.5.

Figure

+ 1

I to I I I)

to (= 0))

xo))

+ 2

I to I I)

of i-periodic

solutions

t)

equations.)

of Eq. (4.4) certain special solutions, namely, 1are of As we shall see shortly, ones, I-periodic great significance. of the solutions nonautonomous role similar periodic equation (4.4) playa x = f(x). Analogous to that of equilibria of the autonomous to equation our presentation in Section 2.4, we begin with the following definition:

qualitative study

the

Definition 4.7. x =

f(t, x) is

solution

A

a

called

t.p(t

t.p(t + T,

moreover,

If,

called the

minimal

solution

T-periodic

+ T,

xo) of a l-periodic

t.p(t, to,

to, xo) =

to, xo)

A solution if is 1 periodic f(t, x) 4.8.

Lemma x = Proof.

then


E for all t The < solution to. to, Yo) to, xo)1 periodic cp(t, if is to be it is not said unstable stable. xo)

solution

ity

Ir.p(t,

r.p( t,

to,

Definition 4.10. be asymptotically Yo satisfying

xol

an

of

to,

Icp(t,

r.

stable

asymptotically

the examples

back to

in the

previous

periodic solution, you section, in particular,

4.4.

Example

begin our investigation For this purpose, we

We now of

such that
0, to, xo)1 -+ 0 as t -+ +00 for cp(t, yo) cp(t, to, xo)

is stable

if it

Iyo

to refer

wish

solution

periodic

illustration

As an may

A stable

of to,

independent all

115)

Equations

on to)

not

and

of Periodic

Geometry

(4.4).

Eq.

that the

the

asymptotic that

suppose

of the solution cp( t, sequence of functions on

behavior

asymptotic

at

looking

the

of will

following [0,

is increasing in U+ and for all t > 0; see

bounded

any solution

u_

O},)

of

period

that

Ic(t)1

satisfies

the

+ M

0 and

{ (t,

A solution remains


O. 0, large, Ixol II(xo) II(xo) the of II must cross the that II has a fixed Therefore, is, diagonal, graph to a solution point corresponding I-periodic (t). We now show the uniqueness of . If we let y(t) - x(t) (t), then Eq. (4.12) becomes) iI

The and are

==

expression

'P

-(

+ y)3

inside

+

3 ==

-y{(

+ y)2

the braces is a positive

y, that is, it is nonnegative If you have no previous zero. +

solution.

a periodic

approaches

and

+

y) +

definite

vanishes

knowledge

( +

of

2}.)

quadratic

form

only when both quadratic

forms,

in

variables

consider)))

126

Scalar

4:

Chapter

Nonautonomous

Equations)

variables inside the braces as a function of two independent the expression that the expression u = 4> + y and v = 4>, and find its zeros. This shows if y f o. This implies that is positive inside the braces y( t) \037 0 as t \037 +00. is stable and every the solution Therefore, I-periodic asymptotically 4>(t) solution of Eq. (4.12) tends to (t) as t \037 +00. Thus, 4>(t) is the unique that the Poincare solution of Eq. (4.12). This result implies map I-periodic of Eq. (4.12) looks like that of Example 4.11; see Figure 4.7.0 qualitatively we investi4.19. In a fluctuating environment: In this example = of the x ax Let a logistic equation gate generalization r(t), k(t) (1 x). and consider functions the equation) be I-periodic, continuous, positive Example

x = r(t) x 1 -

[

can

One

regard

population where

this differential

exhibit periodic Let us first show

k( t)

intrinsic

the

equation as a

this

that

seasonal)

equation

for

model

rate r(t) and

growth

example,

(for

\302\267) k\037t) ]

the

the

of a

growth

carrying

capacity

fluctuations.

has at

least

two

I-periodic

solu-

To solution. The trivial solution x(t) - 0 is obviously a I-periodic if is a solution with that < look for a second observe 0, solution, x(t) x(O) is no I-periodic then x(t) < 0 for all t > 0 and thus there solution with we must show that there is a I-periodic negative initial data. Consequently, < k(t) < KM for initial data. To let Km this solution with end, positive > some constants KM, Km, K M . Then for any solution x(t) satisfying x(t) 0 < x(t) < Km, then we have x(t) < o. Also, if x(t) is a solution satisfying > O. Therefore, any solution x(t) with initial data 0 < x(O) < Km x(t) and approach a for all t E JR, approach zero as t \037 -00, must be bounded this I-periodic solution I-periodic solution (t) as t \037 +00. Furthermore, < 4>(t) < KM. has the property Km It is possible to show that 4>( t) is the only I-periodic solution with initial data; hence, it is asymptotically stable. It is not entirely positive trivial to prove this fact; see the exercises. 0

tions.

We end ble

this section

generalizations

tion with

periodic

examples

=

it,

b,

and

above.

and problems related to possithe Riccati equainstance,

For

coefficients,)

\037\037

where

remarks

some

with

of the

b(r)

+ a(r)x

- c(r)x 2 ,)

with I-periodic functions c(-r) > 0 for 4.19. The form of this equation can be simplified the independent variable T to t through the formula Then the Riccati equation becomes)

c are

continuous,

all

T, generalizes Example if we change somewhat T =

J\037

c-

1

(s)ds.

x =

b(t) +

2 a(t)x - x ,)

(4.13

))))

4.4. where a = ale and in absolute value, periodic solutions

are

see the

such solutions;

the

Consider

x=

3

_x

arguments

of

for

is no

the

I-periodic solutions of number

odd

an

be

The

solutions?

(4.14) is

1.

special

difficult;

Eq.

exercises.)

,.c::>teo)

that the

4.16. Show

If

equation

I-periodic

and e(t) are

d(t),

c(t),

x =

+ c(t)x

2

+ d(t)x

+

show that

the

e(t))

one I-periodic solution. is unAlso, if this I-periodic solution show that there must be another I-periodic solution. Show that x < 0 if x is large enough, and x > 0 if -x is large enough.

Hint:

that

Suppose

I(x)

\037 +00

differential

I is as x

x =

equation

Hint: Use the 4.19. Hyperbolic

Mean

with

function

and

- I(x)

Value

Show that

\037 -00

+ c(t),

where

solution

Equation: the

Riccati

> 0

for

as x

\037 -00.

all

x and Show

satisfying that the

I-periodic c(t) is a continuous which is asymptotically stable.

Theorem. solution

I-periodic

If a(t) and equation)

x=

the

are

b(t)

b(t)

c.p(t,

I-periodic

hyperbolic

solutions, that is, Poincare map are isolated.

Riccati

I' (x)

f(x)

I-periodic

is isolated: A

II' (xo) 1= 1. other I-periodic if

of the

a 01 \037 +00,

has a unique

function,

that

3

functions,

I-periodic

I-periodic

stable.

at least

stable,

4.20.

_x

c(t) is a continuous is asymptotically

where +c(t), solution and

continuous

equation)

has

5

x = -x

has a unique

function,

4.18.

(4.14)

of solutions, but are answer is yes, but a cases are contained in

Exercises)

4.17.

Using

show

can

did not hold and there

uniqueness

If the

there must three I-periodic

(4.14))

t >

for

this

for

constants

d

= d(t),)

that o. However, the argument In fact, we saw in generalization.

is bounded

of Eq.

discussion

two

than

of period

functions

ones given in Example 4.17, one

solutions.

are hyperbolic, then there no more than complete

and

three

sometimes

were

1-

finite

Furthermore,

no more

+ 1)

d(t

c(t),

continuous

arbitrary

longer valid

for c

that

2.1

Section

c(t + 1) =

(4.14)

Eq.

be a

must

of Example 4.18:)

generalization

to the

every solution uniqueness

are

there are

that

prove

large the

where

case

1], there

i=

is very

(4.13)

the exercises.

see

isolated; to

+ c(t),)

where c(t) and d(t) similar

are

Eq.

In the

127)

Equations

exercises.

following

+ d(t)x

is, II'(xo)

[that

hyperbolic

because they even. It is possible

is

number

the

If the initial data for ble. the solution is decreasing.

of them

number

the

=

b

then

01 Periodic

Examples

0, xo) is called hyperbolic solutions are isolated from

corresponding

I-periodic

+ a(t)x

hyperbolic

continuous

- x 2)))

fixed

functions,

points

prove

128

Nonautonomous

Scalar

4:

Chapter

has at most Hint: Suppose

Equations)

solutions. is a solution and introduce the transforI-periodic cp(t) variable of the variable x == cp + y. Then in the new y, the Riccati

mation

equation

two

I-periodic

that

becomes)

-

iJ == c(t)y If we

l

two cases

4.21.

let w

further

==

c(t) dt

Jo

w == -c(t)w l == Jo c(t) dt

0 and

-

a(t)

+ 1.

then

y-l,

i=

==

c(t)

y2,)

0 using

2cp(t).)

the Now, discuss separately the Fredholm Alternative.

that the logistic In Example 4.19, we observed coefficients > 0 and x/k(t)] with I-periodic r(t) solution. Use the suggestion > 0 has as least one positive I-periodic k(t) below to establish that there is exactly one such solution. Hint: Suppose that solutions with I-periodic x(t) and x(t) are two positive x(t) - x(t) > 0 for all t E [0, 1]. Let v(t) == x(t) - x(t). Then show that) Periodic

Equation:

Logistic

equation x

v

==

==

-

r(t)x[l

< r(t)[l -

- x(t)] x(t)/k(t)

[x(t)

x(t)

r(t)

+ r(t)

-

[1

x(t)/k(t)]v

x(t)/k(t)]v)

and) 1

< v(O) exp

v(l) Since

4.22.

is a

x(t)

equal to

1 and

Generalized

< v(O),

v(l)

which

the

Consider

Logistic:

of the

solution

I-periodic

thus

-

r(s)(l

{1

x(s)/k(s))

equation, the

ds

.)

} term

exponential

is

contradiction.

is a

differential

x

equation

ecologically reasonable > 0 and all t, and M

==

x), where

xf(t,

1

in f following > 0 such that f(t, x) < 0 for t, decreasing in x for x x > M and all t. Prove that If x(O) > 0, then x(t) > 0 for all t. (a) l If is a unique positive I-periodic solution (b) Jo f(t, 0) dt > 0, then there to which any other solution with x(O) > 0 approaches. exercise. For further information, Hint: Use the suggestions in the previous see de Mottoni and Schiaffino [1981]. the

satisfies

4.23. Periodic Harvesting:

the

Consider

x =

logistic

[1 where

is a

h(t)

assume that r( t) ally die out.

Show that

As

there

h( t) > 0

so

that

rm

harvesting:)

function. I-periodic the harvested population

and k(t) are continuous, a I-periodic solution x(t), which

property)

, I-periodic

h(t)x,)

r(t)

usual, is

k(t) ]

with

0

Furthermore,

nonnegative,

continuous,

-

equation

-

r(t)x

conditions:

- hM rM

km

< x(t)


0

xo) of

the

this result one would to which is a difficult map object the solution of is available. To circumvent unless Eq. (4.4) compute general in this section we derive for the derivative a formula of the this difficulty, in of solution terms the and the vector Poincare only I-periodic map cp In doing so, we will also discover some other properties of field f (t, x). which are of independent differential interest. equations and

need

x)

If

4.20.

Lemma f(t,

It may

> 1. ifII'(xo) formula for an explicit

unstable

with

cp(O,

cp(t,

0, xo) =

initial-value

following

0,

0, xo))

cp(t,

(t,

differential

a linear

for

=

z(O)

z,)

equation:) (4.15

1,)

is,)

8cp

axo Proof.

then

Xo,

\037\037

that

solution of a i-periodic equation x = is the solution of the 8cp(t, 0, xo)/8xo

is the

xo)

problem

i =

appear

Poincare

the

The

solution

(t, 0, xo) = cp( t,

0, xo)

cp(t, 0, xo)

both sides

Differentiating

chain rule,

of

=

Xo

8 Xo

[ 10

ax

is given +

this

(s, cp(s, 0,

xo))ds ] .

( 4.16))

by)

I(s,

it equation

cp(s,

0,

xo))

with respect

ds.)

to

Xo,

and

using

yields)

t _8cp

{t 8 f

exp

(t, 0, xo) = 1 +

i

0

_8f 8x

(s,

cp(s,

0, xo))

_8cp

8Xo))) (s,

0,

xo) ds.

the

))

130

Nonautonomous

Scalar

4:

Chapter

Now, if we let

z(t)

differential

equation

have the

tion about

consider

if we

following:

==

is, x(t)

0,

cp(t, ==

z(t)

x

==

of the

term

f(t,

0,

cp(t,

the conclusion

Now,

equation

of the lemma

to

==

field

vector

this

of

{I

of

J0

ax

( 4.18))

cp(l, 0,

II(xo)

map,

_ocp

xo).)

(1, 0, xo).

from

follows

(t,

xo)) dt].

Xo yields

8 Xo

0 such that, for IAI < Ao, there dic solu tions \"pI (A, t) and \"p 2(A, x) which have the also that Show -ao/eo and 1P2(0, t) = - J -ao/eo. and \"p2 is are hyperbolic and that \"pI is unstable

I-periodic

JR. Show

JR x

E

Periodic

of Nonhyperbolic

Stability

equation

=

for

0, suppose these values

IAI

each

IAI small.

that there are three of c and d there are

small.

of these x = f(x)

is a unique the same values

0 there with

+

I-periodic Ag(t,

x),

solutions.

where

the function 9

I-periodic with Ig(t, x)1 < 1, and the function f satisfies f(x) = 0 with f' (x) i= 0 for some x. Show that there are constants c > 0 and solution Ao such that, for IAI < AO, the differential equation has a I-periodic show that this is the only I-periodic \"p(A, t) with 1P(0, t) = x. Moreover, within c of x. solution and some the Poincare Hint: Consider above, map. Use the first exercise continuous

is

results

from

and

Chapter

3.)

of Nonhyperbolic Periodic Solutions of a I-periodic seen in Example 4.25, if a I-periodic solution As we have the stability then differential equation type of the (4.4) is not hyperbolic, be determined from the linearization of the Poincare solution cannot map using Theorem 4.22. In this case, we need to compute the higher-order to the of the Poincare map at the fixed derivatives point corresponding how to do this of this section is to show solution. The purpose periodic solution. We first give the in terms of the vector field and the I-periodic of the statement of the main result, and then use it to determine stability the details of in Example solution 4.25. Finally, we present the I\037periodic for some time; however, the not appear The relief sign does the proof. 5.2.

Stability

important ideas such as transformation of variables. solution of Eq. (4.4) and consider cp(t, 0, xo) be a I-periodic about that 0, xo), is,) cp(t, z(t)

contains

proof

Let variation

x(t)

==

cp(t,

0, xo)

+ z(t).)

the

(5.2))))

135)

136

5:

Chapter

Then the

Equations)

of the

terms

several

first

have the

for z will

of Periodic

Bifurcations

variational

differential

(4.17)

equation

form)

z =

b(t)z +

c(t)z2 +

+

d(t)z3

(5.3))

O(z4),)

and the notation O(z4) deare coefficients functions, I-periodic that terms of order z4 and higher; see the Appendix. Notice of the neighborhood of the zero solution of Eq. (5.3) is equivathe study solution lent to the study of the neighborhood of the Now, I-periodic cpo = let us assume that cp is nonhyperbolic, that is, II' (0) 1, or equiva= O. is I-periodic Then the function y(t) lently fo1b(S)ds f\037b(s)ds where the the

notes

t+ 1

To wit, consider + 1) - y(t) = ft b(s) ds. Since the intey(t is invariant over a period of a periodic function under translation, gral t+ 1 = = is ds ds o. Thus, y(t) I-periodic. b(s) ft f01 b(s) in the variational In order to eliminate the linear term equation (5.3), the introduce by I-periodic change of variable u( t) defined in t.

z(t)

The differential

equation (5.3) it =

c(t) and d(t)

where

c(t) = With

this notation,

are

eJo'

we

The

second

2

+

b(s)ds

3

d(t)u

u

variable

new

+ O(u given

2 d(t) = e

will

have

the

(5.5))

),)

by b(s)ds

and

state

our main

theorem.

is a fixed

point of the Poincare equation (4.4). Then

xo

differential

at

the

fixed

point

form)

4

c(t)

of II

derivative

(5.4))

u(t).)

functions

I-periodic

now

ds

b(s)

the

for

c(t)u

Theorem 5.1. Suppose that with Il'(xo) = 1 of a l-periodic (i)

= eJo'

J\037'

d(t).

(5.6))

II

map

is given by

1 II\"

(xo)

c(t) dt

=

21

2co,

the l-periodic the fixed function c(t) is as in Eq. (5.6). Thus, the at solution Xo; hence, 0, xo) of point corresponding periodic cp(t, if is O. unstable Co =1= Eq. (4.4), If Co = 0, then the third derivative of Il at the fixed point Xo is given where

(ii)

by 1

II/\" (xo)

=

61

where d(t) is as in Eq. (5.6). Thus, periodic solution cp(t, corresponding do > 0 and asymptotically stable if

dt

d(t)

the 0, do

6d o,

fixed point at Xo, xo) of Eq. (4.4), is


o.

1

Thus, it

follows

from

solution

cp(t, 0,

1) =

Proof

of Theorem

equation successive

theorem

the cos

21ft

5.1. The

(5.3) to a transformation

differential

above that the nonhyperbolic 4.25 is unstable;

of Example key

idea equation

of variables.

of the

proof is to

with

In the

constant choice

reduce

I-periodic see

Figure the

5.1.

variational

coefficients by using of the

transformations,)))

137)

138

5:

Chapter

of Periodic

Bifurcations

however, care as to have

The transformations must in the transformed equation

taken.

be

must

t so

differential

the

Equations) be

I-periodic

in

variables

be

.

I-periodic.

of the

a consequence

As

ity of 'P, we have

assumption f01

f; b( s) ds is I-periodic.

that

shown

==

b( s )ds

0, the

nonhyperbolicthe

Therefore,

function

is I-periodic. Consequently, the (5.4) exp{f; b( s) ds} in the transformation functions c(t) and d(t) in Eq. (5.4) are also I-periodic. of the one final transformation Now, our objective is to determine with constant coa differential to variables to convert equation Eq. (5.5) The form of Eq. (5.5) suggests that we terms. efficients in the lower-order

transformation)

the

consider

==

u(t)

2

+ (3(t)w

w(t)

(t) +

,(t)w

3

(5.7))

(t),)

functions to and (3(t) and ,(t) are I-periodic where w(t) is the new variable in the formulae we will of notation, be determined. For simplicity below, A few calculations t if there is no danger of confusion. omit the variable for w is given by) show that the differential equation

2{3w +

tV == (1 +

3,W

makes

choice

we

fact, 1)

chose

the solutions ==

of

==

/3

only

are

c

==

0

if

+ f;

(constant)

+

d

function

==

+ O(w 4 )].)

2{3/3)w

3

+ O( w

form:)

4

).

(5.8))

differential

the

that

way

1, the


\" 8w

o

terms in the Taylor

Bo.

(0, 0) of

series

the

(5.21) Poincare

map is

by

n(A,

wo)

==

AOA

+ O(A

+

The ential

AD,

0)

following

equations-is

[co

2 +

) +

[1 +

consequence

+

2 O(A

)]WO (5.22))))

+ O(wg).

O(A)]w6

theorem-saddle-node

an easy

\037BOA

bifurcation of

the

remarks

for I-periodic in Section

differ-

5.1.)

141)

142

5:

Chapter

Bifurcations

5.3.

Theorem ential

of Periodic

Equations)

that

=1= O.

Suppose

AoCo

equation

(i) no l-periodic solution if AAoco (ii) one l-periodic solution if AAoco (iii) two l-periodic solutions if AAoCo

turn to

We now calculating

>

near

A

the

zero,

differ-

In

0; =

0; O.


0, xO center-with E IR? the))) find a IIbifurcation one to IleAtxO IIxo II (9.1)))) attempts point-an organizing

8.1. Properties Definition 8.2.

and x 2 (t) of Eq. (8.2) are said xl(t) I for t each E if, independent IR, the relation C1 x (t) + C2X2 (t) that Cl == 0 and C2 == O.

linearly

implies

solutions

Two

of xl (t) and x 2 2 x 2 matrix whose

Linear independence of the determinant

the

is equivalent to consist

(t)

columns

the

fact

to

be

==

0

that

two

of these

is nonzero:)

vectors

det (x

of

I

2

Ix

( t)

to manipulate a

order

In

bit

of Linear Systems

of Solutions

( t ))

of

pair

all t

for

\037 0)

solutions

E

IR.)

effectively

(8.3))

we introduce a

terminology.

and x 2 (t)

two solutions of Eq. (8.2), then the whose columns are the two solutions (t)), of Eq. =1= 0 for all (8.2). If, in addition, det X(t) t E JR, then X(t) is said to be a fundamental matrix solution of Eq. (8.2). == A special fundamental matrix solution satisfying the condition I, X(O) where I is the 2 x 2 identity matrix, is called a principal matrix solution. 8.3.

Definition

Ifxl(t)

matrix X(t) (Xl(t) is called a matrix solution 2 x 2

I x

are

2

a lemma which provides the useful fact that it suffices at only one value of t, and gives an explicit formula the flow of Eq. (8.2) in terms of any fundamental matrix solution. state

now

We

to

check

det X(t)

Lemma 8.4. Properties of fundamental solutions: of Eq. (i) If X(t) is a matrix solution (8.2) with det X(O) det X( t) \037 0 for all t E IR, that is, X( t) is a fundamental of Eq. (8.2). (ii) IfX(t) is a fundamental the

satisfying

matrix

cp(t,

==

xO)

then

solution,

condition x(O)

initial

X(

== XO

-1

then

solution

the solution of Eq.

is given

t) [X(O)]

\037 0,

for

(8.2)

by)

xO.)

(8.4))

of the initial-value problem x(O) == (i) First observe that the solution == o. is zero: that there Now, Eq. (8.2) identically suppose c.p(t, 0) == are T constants and such that where + Cl, C2, 0, CIXI(T) C2x2(T) xl(t) and x 2 (t) are the columns of X(t). Then, CIXI(t + T) + C2X2(t + T) is also a solution for Eq. (8.2), which for t == 0 is zero. Therefore, by uniqueness

Proof. for

o

of

we have)

solutions,

cp ( t,

Thus,

if we

take t

==

0)

-T,

==

then

o == cp( -T,

Now, the Cl

linear

== C2 == O.)))

independence

Cl X

I

we 0)

(t + T) +

C2X2

(t + T).)

obtain)

==

of the

C1XI(0)

+ C2X2(0).)

vectors xl(O) and

x 2 (0)

implies

that

219)

220

Linear

8:

Chapter

(ii) The

cause it

right-hand

X(O) X(O)-l = I,

it is

is a solution of Eq. (8.2) besolutions x 1(O) and x 2 (O). Since initial condition. Thus, from the

(8.4)

Eq.

of the the

satisfies

also

it

theorem,

uniqueness

of

side

combination

linear

a

is

Systems)

the solution. 0

The of superposition principle implies that the set of all solutions The lemma above shows that the dimension Eq. (8.2) is a vector space. of this vector space is two. After some general facts about the discovering we will the flows of linear determine bases for vector explicit space systems, of solutions of Eq. (8.2) for any given coefficient matrix A. in the first we saw As of our book, the flow of the scalar example linear differential equation x = ax is given by the exponential function = eatxo. To obtain an analogous formula for the flow of linear c.p(t, xo)

planar systems we

-

eAt where for

the

X(t) is any fundamental flow of Eq. (8.2)

the

hence establishing

notation)

the

introduce

be

written

c.p(t,

Xo)

desired

and thus eAt

8.5.

Lemma

(8.6))

Of course,)

AO

the reason

eAt and provide

solution

The

=

for

I,)

choice

this

(8.2). of the principal of notation.

eAt satisfies

solution

matrix

principal

Eq. (8.4)

= eAtxO,)

is a principal matrix solution of Eq. collect several important properties

now

We

ofEq.

as the

analogy.

e

(8.2). Then, matrix exponential)

solution

matrix

can

(8.5))

X(t)X(O)-I,)

matrix

the

following

properties:

(i)

= eAt e As

eA(t+s) -1

(ii) (iii)

= e-

(eAt) :t eAt

( iv ) eAt

= (i)

Proof.

= Ae

For Eq.

implies the

desired

Take

(iii) The

;

=

(8.2).)))

n =

fixed s,

s =

2 2 I + At + l.A 2!. t + ...

the

which

(8.2)

matrices

coincide at t

eA(t+s)

=

o.

and The

eAt e As

are matrix

uniqueness

theorem

equality.

-t

first

Eq. (8.5). The second rem with the observation of Eq.

eAt A;

lAnt n!

any

solutions of (ii)

At

At

\",+00 L....in=O

;

in

property

equality one

(i). follows from

is again that both

the definition a consequence of the Ae At and eAt A are

of

eAt

uniqueness matrix

given

in

theosol\037tions

8.1. Properties (iv) This ones

previous matical

more

is considerably

property

a complete

and

difficult to establish than the a certain amount of mathe-

present the

we will

Here,

sophistication.

requires

proof

of Linear Systems

of Solutions

essential

steps

to make

it

convIncIng.

We should

that it is possible to take property (iv) as the defithat one establishes the of this matrix provided convergence all series. one can demonstrate the other of eAt power Then, properties that eAt the matrix solution of the result is principal Eq. including (8.2). we will establish the however, Following our presentation in this chapter, for its power series expansion from the fact that eAt is the principal formula matrix solution of Eq. (8.2). eAt. observe For the sake of brevity of notation, let us set P(t) Now, the that for each vector XO E JR, the matrix P (t) satisfies equation integral remark

of eAt

nition

P(t)xO = Since P(t) iteratively

Ixo +

it

ds.)

AP(s)xO

(8.7))

appears on both sides of this equation, we attempt to find P(t) If we take as our initial by using successive approximations.

guess) ==

p(O)(t)XO

and

compute

the successive iterates = Ixo

p(kH)(t)XO

as the

then

kth

for

iterate,

k

==

Ixo,)

with)

+

(s)X

it

1, 2, . .

0,

O

AP(k)

., we obtain

ds,)

(8.8))

the

ex-

polynomial

preSSIon)

p(k)(t)xO

1

+ AxOt

== Ixo

+ ,A2xOt 2.

2

1 +...

+

k ,.)

AkxOt

k

.

of vectors given by this p(k)(t)xO first observation is that it suffices to +00. converges of p(k)(t)xO for all XO on the unit circle 8 1 show the convergence {x : == in as can be written a scalar because vector JR2 multiple any II xII I} only, 1 to the is realization a study of point the fact that,thatsince of some vector on 8 . The second observation IIAxol1 will not beand trivial. difficulties associequilibria nonhyperbolic XO a closed thebounded is continuous 8 1 isDespite as a function of set, there with ated role them, however, equilibria playa prominent Q > 0 suchnonhyperbolic that) exists a constant in our subject, as we shall soon see.) 1 for all xO E 8 .) < Q) IIAxOl1 We

now

show

that

as k

formula

Exercises)

the

---+

sequence

The

.t.Q.)

it is now easy to establish the estimates following of the following scalar 1.8. Many examples: Determine the equilibrium points p(k) (t)xo: on the norms of the iterates differential and compute the linear variational about equations equations the equilibria. 1 2 2equilibria 1 the hyperbolic and their Identify stability types. k < 1 + at + eat))) a t + . .also. + akt

for

then there

O.)

in the next section.

apparent

initial-value merical algorithms for solving For example, try to solve the Euler, Runge-Kutta, and various Improved

equations.

parts,

x E ]R?,)

all

I/xll)

system: Since the exact serve down, they often

real

negative

at

become

will

A have that, for

of linear systems can grounds for various nuproblems for ordinary differential linear system using Euler, following solutions

as testing

sizes:)

step 1998.0

-1999.0 )

Compute the eigenvalues explicit solutions; compare

x.)

and

the

your

theoretical

eigenvectors findings

of the with

systhe)))

8.3. numerical

ones.

mathematical

esting things such a behavior and wait, or step

size

ear

system

more

drastic

really have a precise are doing inter-

time

scale.

small\"

To

capture

size, and vary the the eigenvalues of the linon this linear system, see \"very

changes

of

information

Systems

solutions

as on a long, must use either a

places where the How do the magnitudes

above compare? For et al. [1977], p. 124.)

not

well

one

the

accordingly.

Forsythe

short, as

numerically, detect

term stiff does means that the

way, the it usually

the

By

definition; on a very

in Linear

Equivalence

Qualitative

step

are

in Linear Systems Equivalence The of this section is to investigate the question of qualitative purpose of planar linear in the spirit of Section 2.6. Let equivalence systems of qualitative equivalence. begin with a precise definition of the notion 8.3.

Qualitative

us

linear systems i = Ax and i = Bx are said if there is a homeomorphism h : IR 2 \037 IR 2 equivalent topologically of the plane, that is, h is continuous with continuous inverse, that maps the orbits ofi = Bx and preserves the sense of the orbits ofi = Ax onto

Definition 8.13. to

direction of Since

convenient by

Two

planar

be

time.)

have a formula to recast this definition

we

one flow to

mapping

for

the other, h(eAtx)

flows

the in that

of planar

a somewhat

linear systems,

more quantitative

it

is

form

is,)

= eBth(x))

(8.20 ))

for every t E IR and x E ]R2. A homeomorphism h satisfying Eq. (8.20) in Definition 8.13; while than it the one required is a bit more special orbits onto orbits, it also preserves the time parametrizations of the maps orbits. However, as we shall shortly see, in the case A and B are hyperbolic, nonzero real parts, the homeomorphism h in chosen to satisfy Eq. (8.20). of linear is a someThe question of topological equivalence systems to motivate the introduction of the formal what difficult one. Therefore, let us first reexamine the redefinition of qualitative equivalence above, of transforming matrices into sults of the previous section. In the course their Jordan Normal Forms we have investigated the question of topological in a limited context by considering linear maps only invertible equivalence and In as our allowable fact, Eqs. (8.9) homeomorphisms. imply (8.11) A = P-1BP, that if there is an invertible 2 x 2 matrix P such that then the flows of i = A x and i = B x are related by PeAt = eBtp. In other if matrices A and B are similar, also called conjugate, then the flows words, i = B x are the))) of i = A x and It is easy to verify linearly equivalent.

that

is, their

Definition

eigenvalues

8.13

have

can indeed be

237)

238

Linear

8:

Chapter

Systems)

converse implication by simply and then evaluating it at t = o.

the

differentiating

= eBtp

PeAt

equation

is a bit too restrictive for comparing Unfortunately, linear equivalence features of flows of linear For the two qualitative example, systems. x = - 2 I should be linear systems x = - I and considered qualitatively this follows from the fact equivalent equivalent; yet they are not linearly that the matrices -land - 21 are not similar because they have different as shown in the exercises. eigenvalues, the differenone may naturally ponder After about linear equivalence, A tiable equivalence of linear For two matrices and B, does systems: given there a diffeomorphism h : IR 2 \037 IR 2 satisfying h(eAtx) = eBth(x)? exist In the case of hyperbolic linear systems, differentiable equivalence offers

the

as the

new,

nothing

8.14. Two

Lemma

Linear

Proof.

we

therefore,

linear

hyperbolic

equivalence,

of

converse

Ax and x = Bx

are

equivalent.)

linearly

differentiable

implies

course,

to show the

need

x =

systems

only if they are

if and

equivalent

differentiably

attests:)

lemma

simple

following

Suppose

implication.

equivalence; ........ that h :

2

is a diffeomorphism satisfying Let h(O) = c. Since Eq. (8.20). c is an equilibrium that equilibrium point of x = Ax, it follows the diffeomorphism of JR? point of x = Bx, that is, Bc = o. Now, consider of the shift g(x) \037 x-c. takes orbits The diffeomorphism consisting 9 ........ of x == Bx into itself, and the diffeomorphism h = 9 0 h takes orbits of x = Ax to the orbits x = Bx that while leaving the origin fixed, is, h satisfies Eq. (8.20) and h(O) = o. with to xO and set xO = 0, Now, if we differentiate Eq. (8.20) respect we obtain HeAt = eBt H, where H = Dxh(O) which is a linear map. If we to t and set t = 0, then HA = BH. differentiate with respect )

IR 2

o

\037 IR

is an

It

but

is evident

face

the

difficult

from the foregoing task of deciding

systems using homeomorphisms Let us now state two theorems lengths,

systems and discuss we will defer the proofs

theorem

covering the

nar

linear

Theorem have

nonzero

8.15. real

hyperbolic

of

equivalence

qualitative

choice

of linear

plane.

topological classification of plaimplications. Because of their end of the section. Here is the first

on the

of their

some to

the

linear

Suppose that the Then the

parts.

the

we have no

that

discussion

the

systems.) eigenvalues

two

linear

of two systems

A and B Ax and x = Bx

matrices

x =

if and only if A and B have the same number equivalent topologically of eigenvalues with negative (and hence positive) real parts. Consequently, there are three distinct equivalence classes up to topological equivalence, of hyperbolic linear the following repreplanar systems with, for example, are

sentatives:)))

8.3. two

(i) (ii)

books

elementary of

equivalence

and one negative

positive

(\037 \0371):

In

eigenvalues;

positive

one

(iii)

ear

two

(\037 \037):

Systems

eigenvalues;

negative

\0371):

(\0371

in Linear

Equivalence

Qualitative

on differential linear

hyperbolic

eigenvalue.

equations, most

often

only

is considered and a

systems

the lin-

host

of

are introduced to label improper node, spiral, etc., various see Figure 8.4. From the topological phase portraits; viewpoint, there ar\037' only three cases and they are determined however, solely by the in the theorem above. signs of the real parts of the eigenvalues, as asserted Notice in particular the striking assertion that a stable spiral and a stable node are topologically equivalent. Consequently, it is preferable to emthe whose ploy following terminology appropriateness will become apparent when we study the qualitative features of nonlinear differential equations near an equilibrium: A linear system whose have negative real eigenvalues is a called a linear whose have parts sink; hyperbolic system eigenvalues

as node,

such

terms,

a hyperbolic positive real parts is called and the other is the negative positive

and, when one system is said to be

source; linear

is

eigenvalue

a hyperbolic

saddle.

We next

the

present

of nonhyperbolic

classification

topological

linear

systems.

If a coefficient then the planar

8.16.

Theorem zero

real

part,

equivalent to precisely indicated coefficient

matrices:

zero

matrix;

( i)

(ii) (iii) (iv)

(v)

(\037\037):

(\0371

\037):

(\037\037):

(g

the

\037):

one

negative

one

positive

two

zero

two (\0371

one

\037):

of

linear

the

and one and one

eigenvalues

purely

imaginary

x

system

following

zero

zero

at least one

A has

matrix

five

==

linear

eigenvalue

with

is topologically with the systems

A x

eigenvalue;

eigenvalue;

but one

eigenvector;

eigenvalues.

Phase portraits of the representatives and of the three hyperbolic the linear systems are depicted in Figure 8.5. From a visual inspection of phase portraits of linear systems, the conclusions of the two theorems above are quite plausible, yet formal proofs turn out to be somewhat long and intricate. We now this arduous begin will which our attention the of this task, occupy part during remaining with some results on forms. section, auxiliary quadratic A real symmetric matrix C, that is, C T == C where the superscript T denotes the is said to be positive if the quadratic form))) transpose, definite five nonhyperbolic

239)

240

Linear

8:

Chapter

Systems)

about the

Information

Stewart the

subject

erence Milnor

and

[1963]. of

application

is a de

case

not

continuum

Melo

[1982].

are not isolated single point; see, for Henry

further

for

[1983]

for our purposes

Theorem

ref-

standard

of Sard

Theorem

is on

is

given

details. 37

page

in

A of

elements it is possible

of the set that

of

the

of equilibria; such an example are several applications and yet the w-limit set

is

There

Aulbach

example,

of a gradient set of a bounded on page 14 of Palis

equilibria

w-limit

where of

a

the

bounded

[1984], Hale and

equilibrium orbit is

Massatt

a

[1982],

[1981].

Gradient catastrophe

the

where

isolated,

points and

Sard's

the

and

Appendix

of the

statement

The

points

role throughout this chapof a real-valued function

[1963].

In the system are orbit

played

a prominent

consult Milnor [1965] or Smith

the Appendix; Milnor

@)'@)

functions

of

in

gradient.)

study of nondegenerate critical of Morse see the Theory;

A deep

relevant

name

umbilic is stored

elliptic

Notes)

Critical points is the

under the

of the

field

in Poston and

is contained

umbilic

elliptic

vector

gradient

of PHASER

library

Bibliographical

ter.

The

[1978].

systems have theory; Zeeman

see, [1977]. for

diverse

example,

uses.

play They Thorn [1969],

an important Poston and

role

in

Stewart

in Morse In differential topology, especially to anone flows vector fields to take one manifold along gradient theory, as in are in described Milnor Similar ideas used Smale other, [1963]. [1961 in higher In and to affirm the Poincare conjecture dimensions. 1961a] numerical analysis, computing methods under the names \"conjugate graor descent\" essentially consist of flowing dient\" \"steepest along gradient vector fields; see, for example, Conte and deBoor [1972]. Because of the an equilibrium, fact that bounded solutions approach computations yield convergent results. The of a vibrating membrane is studied in Chow, Hale, and example Mallet-Paret is for reaction-diffusion equations [1976]. A good reference Fife [1979]. It is evident that the dynamics of a gradient are essentially system determined by the equilibria and the possible orbits between connecting of This observation can be made and equilibria. precise practical any pair combinatorics. the vertices of a graph One associates by resorting to simple with equilibria and the edges with orbits. Such graphs are used connecting 8.4. All linear whose have all two-dimensional to Peixoto in as explained Figure systems eigenvalues negativeby real (a) gradient flows, classify Hale Peixoto [1973] andtopologically parts-sinks-are equivalent (continued).))) [1977]. It is not possible to characterize all structurally stable systems in dithan two. However, there is a nice result in Smale [1961]))) mensions greater [1978],

and

8.3.

Figure 8.4 Continued. x

T Cx

> 0

for

all

x

(b)

=1= o.

form on

definite

An

saddles

Linear

important

the

k >

any

geometric property of a positive the level set { x : x T Cx = k }, In the lemma below, we origin. with

forms

of

additional

properties

8.15.

Theorem

there

then

a positive

exists Proof.

Without

loss

Normal

Form

Jordan

Case

equivalent.)

topologically

of A have negative real parts, 8.17. If the eigenvalues definite matrix e such that ATe + CA = -I.

Lemma

Case

are

Systems

is that

quadratic plane the 0, is an ellipse encircling establish the existence of some quadratic which will playa role in the proof crucial for

in Linear

Equivalence

Qualitative

(i):

(ii):

A =

(-;'

A = (::::\037

of generality, we may assume and consider three cases: Al -\0372)'

!a)'

0: >

> 0,

A2

0, and

>

O. Take

that

the

f:. O. Take

is in

C = (l/(\037A.)

(3

A

matrix

C =

1/(20:)

l/(\037A2\302\273)'

I.)))

241)

242

8:

Chapter

Linear

Systems)

x2

X

X

2

2

\037

,J

x,

X,

x,

(

\\ -1)0)

(-1 o

x

. . . .

(\037

(\037

-

-

..

.... . . ...

. . .

x 2)

x 2)

2)

.

-)

g))

(-6

(\037

. -

\037 ..

- - - ..

\037

(g

\037))

x 2)

x 2)

-- -

x,

x,

..)

6))

8.5. Phase portraits of classes of planar linear systems.)))

Figure

-

-

..

-

x,

x,

\037))

-

-\037)

(\037

\037)

(-\037

representatives

6))

of topological

equivalence

x

8.3. Case (iii): A =

1/(2A)

c > 0,

any

-1

A TD + DA

c/(2A)

==

-1) )

( cj(2A))

x

\037

l / 2x.

E-

If we

(\037 \037 )x.

A =

let D

=

-E.

introduce the change l / 2 DE- l / 2 possesses

0 and

>

E-

matrix

C

first

main

the proof of our

turn

now

of

co-

the

result

of the

section.

Let us first observe 8.15. that the conditions on the A and B are necessary for topological For equivalence. if a homeomorphism is not difficult to persuade oneself that

Theorem

of

Proof

that

0

properties.

We

1 - [c/(2A)]2 Then the

that

c so

choose

ordinates

desired

I--->

Systems

then)

I,

Now,

x

assume

can

we

of coordinates

transformation

the

by

\037>.)

(-0>'

For (\037O>' \037\\).

in Linear

Equivalence

Qualitative

matrices

coefficient

this purpose, takes a bounded

it

A x to a bounded positive orbit the w-limit set of one the homeomorphism also takes orbit to that of the other. A similar remark holds for a-limit sets positive of bounded orbits also. Consequently, if A has both eigenvalues negative real and A is topologically equivalent to B, then with negative parts every of x == B x approaches zero as t ----* +00. solution both eigenvalues Thus, of B must have negative real parts. A similar argument holds, as t ----* -00, If A if both eigenvalues of A have has one real and positive parts. positive of

==

x

must eigenvalue, then there zero as t ----* approaches

as t

zero

approaches

one

that

B x

==

x

==

then

B x,

one negative of

of x

orbit

positive

----* -00.

one nonequilibrium solution and another solution that

be +00

B must

Consequently,

have one negative and

eigenvalue.

positive

To prove

sufficiency, let us

begin

with

Case

(iii)

as it is the simplest. B have one linear change

we may assume that if A and section, previous and one positive eigenvalue, then they can be put, by negative of coordinates, into the following Jordan Normal Forms:) the

From

Ai > 0

each

linear

scalar

and

differential

J-li

topologically

ilarly, the

scalar

B==

(

\0372) >

O.

Now,

J-ll

o)

,)

:2)

from Section 2.6

recall

that

the

two

: JR ----* JR.

Sim-

equations) ==

Xl

are

,)

o)

( where

-

-AI

A==

equivalent

-AlXl,)

with some

Xl

==

-

J-ll Xl)

homeomorphismhI

equations)

X2 == A2 X 2,)

X2

==

J-l2 X 2)))

243)

244

Linear

8:

Chapter

Systems)

are topologically equivalent via a homeomorphism h2 : of the planar linear x = A x and x = B x are systems with the equivalent homeomorphism) h : JR2 ---+ JR2;

h(x)

= (hI

JR

The

---+ JR.

then

flows

topologically

(Xl), h2 (X2)).)

and construct homeomorphisms to establish of planar linear whose have systems eigenvalues t Case negative real parts. replacing (ii) follows from Case (i) by simply but the with -to The construction below a bit technical, may appear idea behind it is quite simple. For each linear system, find an geometric the that the such each orbit, equilibrium ellipse encircling origin except the ellipse in the same direction and only at one crosses point at the origin, and point. Now, map homeomorphic ally one ellipse to the other, map an orbit through a point on the first ellipse to the orbit passing through the on the second ellipse. extend such a homeomorphism point image Finally, to include the This idea is reminiscent of polar where coordinates origin. the ellipse is the \"angular variable\" and the orbits are the \"radial variable.\" To prove the existence of two ellipses with the desired properties mentioned some ideas from the theory of \"Liabove, we use Lemma 8.17 and functions.\" We will delve into the topic of Liapunov functions in apunov 9. Here, we will be content to point out that if the derivative of a Chapter

now turn

We

the

to

Case

(i)

equivalence

topological

of a linear positive definite quadratic function along the solutions system x = A x is negative, at the then the orbits cross the elliptical except origin, level sets inward. There exist two positive definite symmetric matrices CA and C B such that)

+ CAA

ATCA

Let x( t)

be

d

dt [xT

a solution

(t) C A

x(t)]

of x

= -I,)

= A x,

= xT (t)

BTC B

+

[A TCA + CAA] x(t)

=

_x

T

(t) x(t).)

A x crosses the level sets that any nonequilibrium x T CBx inward see also;

8.6.

a homeomorphism h : JR2 = A x to the orbits of x = B x. Let T I} ---+ {x : x CBx = I} be a given homeomorphism For any xO i= 0, there is a unique time txo such that xl = 1. Now, define the ellipse XTCAx h(xO) = eBtxOh(xl); continuous It is evident, from of solutions dependence in fact, it is a diffeomorphism h is a homeomorphism;

It is

takes

= -I.)

then)

Therefore, any nonequilibrium solution x( t) of x = A similar of x T C AX inward. computation shows solution x( t) of x = B x crosses level sets of the Figure

CBB

the

now

easy

orbits

a diffeomorphism.)))

to define

of x

- 0

---+ JR2

- 0

that

= of the two ellipses. e-AtxoxO lies on Ii : {x

see

: XTCAX

Figure

8.7.

on -.initial data, that if h is chosen to be

8.3.

in Linear

Equivalence

Qualitative

x 2)

Systems

x 2)

x 1)

x 1)

x =

x = Ax)

8.6.

Figure

linear

ative

quadratic

Orbits of

a planar

parts cross the function, ellipses,

linear

system of an

curves

level

Bx)

whose eigenvalues have negdefinite appropriate positive

inward.)

h(x 1 ))

Figure

nar

a

8.7. Constructjng linear systems whose

eigenvalues

have

negative

real

8}. Since

Then

extend

there

the set

is to T {x : x

==

with

to(8)

CBx

==

I}

txo > to, is bounded

two

pla-

parts.)

the domain of the homeomorphism h above to show that h is continuous we define h(O) == O. It remains xO E and consider purpose, suppose that 0 < 8 < 1 is given To

of

the flows

between

homeomorphism

to at

of R 2 , o. For this all

{x : XTCAx < and t o (8) ---+ +00 as 8 ---+ O. by a positive constant, say,)))

245)

246

Linear

8:

Chapter

M, and

Systems)
0 and

some

a, it

k and

constants

positive

that)

follows

Ilh(xO)11

< ke- ato

= IleBtxOh(e-AtxoxO)II

(6)M.)

so long as is 8 > 0 such that Ilh(xO)1I for any \342\202\254 < \342\202\254 > 0, there is norm equivalent to the Euclidean norm, (xO)TCAXO < 8. Since XTCAx h h is continuous at the origin. As we saw in an example above, however, a diffeomorphism at the origin. ) /\302\273){).,

!::,i!\"!!:::f::!:':!! .\".\" \". ...................) ...................... . ....................... ....................... ...............)

.:.:::.:.:.::::::::::::.:....... \" ,) \0370\037jliillli)

Poincare map

8.11.

Figure

The behavior of of the iterates of the

of Eq.

solutions

C : ill?

(8.32),

see

you

linear will

Poincare

undertake a detailed

that

its stability

generalized to

are easily

the

dynamics

map) xO

\037 C

of the

map

of planar

study

ones, in Chapter 15. For find the few remarks below

and

point

xO

ill?;)

in

reflected

,)

linear system

I-periodic

8.11.

Figure

We will the

as the

be viewed

should

which

\037

(8.32) are linear

two-dimensional

linear system.)

a l-periodic

of

have

we planar

the

moment,

acceptable.

developed This maps.

maps, including, of course, however, we trust that The notions of a fixed

3 for

in Chapter essentially

scalar

maps

entails replacing in several norms, Definitions 4.9 and

scalar with and absolute values with quantities vectors, definitions in Chapter 3. With similar replacements, of stability, 4.10 are also readily generalized to yield definitions asymptotic of the I-periodic system (8.32). stability, and instability of a solution A periodic to a fixed point of the solution of Eq. (8.32) corresponds linear and the of the planar map periodic solution of stability type (8.34), is of the fixed the same as the point Eq. (8.32) stability type corresponding in particular of the map (8.34). Notice that the zero solution of Eq. (8.32) to the fixed the of C at the origin. We now summarize point corresponds main implications of these remarks for the zero solution of Eq. (8.32).

Lemma in Eq. (i)

If

Let Then (8.33). 8.22.

IJ.-lil

1, for i


0 such that) Ilg(y)

Returning

to the

II


0energy f is topologis a neighborhood ofx in which then there point ofx satisfied. from of the in Theorem the estimate invariant (9.1) curves. of more 9.3, we have) integrationThen, to the linear vector field x = Df(x)x. () ically equivalent numerical evidence above, the important question Despite the strong t have a first inof point Henon-Heiles Does the Hamiltonian remains: still of the Because of the hyperbolicity x, the homeomorequilibrium (18.4) < Ke-atllyOIl + ds,))) Ke-a(t-s)mlly(s)11 The answer))) h abovethat can isbeIly(t)1I chosen so as independent to preserve parametrization))) phism tegral functionally l alsofrom thethetimeHamiltonian? yet)

+

l

9.1. Asymptotic as long as Ily( s) II inequality with eat

eat

II


(J.L2 - 4m )y\037 > o. origin. In the region in Suppose that yO E n n U. Then the solution through yO remains n as long as y(t) E U. To finish the proof, we observe that the solution V

( C1)

if C2

let

norm has for some value of t, that is, the solution equal to \342\202\254 hit the boundary of U. This follows because there is 6 > 0 such that The of the instability of the proof V(y) > 6 if y E U and V(y) > V(yO). is now complete. 0 equilibrium point of Eq. (9.12) at the origin

through yO must

Example 9.8. bility

type

of the

The

damped

continued:

pendulum

equilibrium points

(n7r,

0)

of Eq.

the n is an

Let us consider (9.8) when

staodd)))

274

Near

9:

Chapter

Equilibria)

Y2)

C, >

C2 >

=

Y2

0)

-Y,)

Y,)

and the

oEV

sets

level

n

region

of Eq. (9.8) at these

linearization

The

integer.

The

9.3.

Figure

where

V(y) >

points

equilibrium

o.)

is the

linear system)

( The

ejgenvalues

of this

linear

J.11,2

one

Since

Theorem

1

0

x. =

w

2

system

= -a:i:

is positive (and eigenvalue 9.7 that these equilibrium

_ 2a can

easily

2 v a

the

x.)

+w

other

points

(9.13))

)

are

be found to

be)

2 .)

is negative),

it

follows

frorn

unstable.

of the we have plotted in Figure 9.4 the phase portraits these unstable one of nonlinear system (9.8) and its linearization near (9.13) is a saddle. Furthermore, equilibrium points. Of course, the linear system is the nonlinear looks much like a saddle also, and the linearization system of the nonlinear near a good reflection of the local phase portrait system of the The preservation of a saddle under small equilibrium. perturbations a linear is true in general and we will explore this fact further in system For comparison,

Section

9.5.

0)

equilibrium point of a nonlinear system cannot linearization. It is evident from Theorems 9.5 always the of such and 9.7 that a situation can occur only if some eigenvalue has linearization has zero real part (and the remaining eigenvalue negative we must examine nonlinear real part). In this case, effects of the specific \\ve terms of the vector field to determine the local dynamics. Indeed, in the saddle-node))) encountered an instance of this difficulty have already

The stability be

determined

type

of

an

from

9.2.

275)

Linearization

from

Instability

/

/')

its

Local phase portraits of the damped near one of the unstable equilibria.)

9.4.

Figure

linearization

bifurcation

9.9.

Example

When

X2

+

X2 = -Xl

of

the

A is

a scalar

harmonic

linear terms. For

was present.

eigenvalue

suffice:

the

Consider

system

equations)

Xl =

where

and

pendulum

eigenvalues:)

does not

linearization

differential

zero

one

where

with purely imaginary

example

of nonlinear

7.23

in Example

given

is an

Here

nonlinear

parameter.

oscillator all

values

AXI (XI

+

These

x\037)

(9.14)) +

AX2(xI

equations

(7.4) but the of A,

+

x\037),)

are a nonlinear does

perturbation

the origin is an

equilibrium

perturbation

not point

affect

the

and

the)))

276

Near

9:

Chapter

linearized

Equilibria)

equation at

the

is, of

origin

course, the harmonic

0

. x =

oscillator)

1

_lOx.))

(

(9.15))

linear system are :f:i, which have zero real parts. we compute of the nonlinear system (9.14), analyze of a solution from the the derivative of the square of the distance origin:) The

eigenvalues

of this

To

the

behavior

d

dt ,\\ < 0, then Ilx(t)112 the equilibrium at the then all solutions of Eq. in this case, Therefore,

If

2

2 =

+

(Xl

2 2 2 ( xl + X2 )

.)

zero

monotonically stable. asymptotically

approaches is

origin

2,\\

X2 )

initial data

with

(9.14)

the origin is unstable.

xO

i=

as t

--+

+00.

'-'Q.o)

9.5. Equilibrium

Consider the

Pol:

der

Van

of

oscillator)

= X2 -

Xl

X2 =

where

A

is a

the

Show that

f(x)

(c) f(x)

Xl))

type

of the

that

the

as a

differential

only equilibrium

function equations

of

point

is at

A.

whose

vector

fields

Xl

=

(

)

+ 5X2

5Xl

+

XI;2 + X2 - X2)

;

)

eXl +X2 - 1

=

.) sin(xl

all equilibrium and determine their - XlX2, (a) Xl = 1 = 2Xl - xI (b) Xl

Find

;

(Xl :22x\037

( 9.7.

der Pol's

below is unstable:

given

(a) f(x) =

(b)

Show

stability

solution

zero

A(xr/3 -

of Van

form

Lienard

-Xl,)

scalar parameter.

Determine its

the origin.

are

0,

0)

Exercises)

9.6.

Thus,

However, if ,\\ > 0 escape to infinity.

+ X2) points stability

X2 = XlX2,

) of the

following systems

of

differential

equations

properties:

Xl -

x\037;

X2 =

-X2 +

XlX2;

(c)Y+iJ+y3=O; Xl = sin(xl + X2), X2 = e 1; = = Xl X2 2X2 - x\037 - x2xf. x\037 XlX\037, (e) Xl of the first three equations. Make sure portraits Try to sketch the phase at each equilibrium to use the information about the linearized equations

(d) Xl

point.)))

9.3.

277)

Functions

Liapunov

9.8. If the origin is a stable but not asymptotically stable equilibrium the planar system x == f(x), can the origin be a saddle point of the

point

of

linearized

equations?)

9.9.

Feedback control:

for the pendulum of length l, mass of the proportional to the velocity in the pendulum pendulum. Suppose now that the objective is to stabilize the vertical position mechanism its a control which can (above pivot) by move the pivot of the pendulum horizontally. Let us assume that f) is the in the clockwise measured direction and th.e angle from the vertical position to linear force v due the control mechanism is a function of f) and f), restoring that is, v(O, 0) == Clf) + c 2 0. Convince yourself that the differential equation)

m, in a

.

..

0+

0) asymptotically

(0,

equation

friction

\"Pole\" placement:

9 sin 0 1:

-

such

of

in such a

be chosen

C2 can

9.10.

mf)

the motion

describes

the

with

Consider

medium

viscous

way

.

1

-

+ C2f))

1:(Clf)

a pendulum. Show so as to make the

cos

f) == 0)

that the

Cl and

constants

point

equilibrium

(f), 9)

==

stable.)

In the

the

above,

problem

linearized

equations had the

form)

x == Ax

+

by,)

b=(\037),)

and

the problem was

the eigenvalues If the

Theorem: is

matrix

matrix

controllable,

A + beT

have

Hint: There

9.3.

the

T

x == ClXl + C2X2 so that A + beT have negative real parts. Prove the

by

choosing

v

==

e

result:

following

above

of

solved

is

negative a vector

(b

then

is nonsingular, that is, the linear system I Ab) there is a vector e such that the eigenvalues of

real parts. e with tr (A

+ beT)

< 0

and det(A

+ beT) >

o.)

Functions

Liapunov

x of a planar system x = For an asymptotically stable equilibrium point estimates it is of considerable practical importance to obtain good f(x), of the the subset of ]R2 of the basin attraction of that x, is, consisting of initial data XO with the property o. The projection of these level in concentric ovals encircling the origin;

sets onto see

Figand

of course, obvious in the examples above, for certain classes of functions known locally

functions\"; namely, those 2 ues of the Hessian matrix (8 V(x)j In this case, the positive. Implicit \"Morse

2

functions

8x i 8xj)

Function

V (x)

for which

evaluated at Theorem

local

the

as

eigenval-

minima

implies that

are these)))

9.3.

279)

Functions

Liapunov

v)

x 2)

V(x 1 ' X 2) =

= V(x 1 ' X 2)

c 2)

c2

= V(x 1 ' X 2)

c 1) x 1)

X 2)

x,)

the

Graph and

9.5.

Figure

Also, one can

are isolated.

minima

local

of a

curves

level

positive

V near

function

definite

origin.)

the minima are diffeomorphic isfactory characterization In

homogeneous

Here is an

variables.

of

level sets of

choice of positive

standard

the

situations,

simple

from

come

the

V near a sat-

However,

positive

definite

available.

is not

function

that

show

to circles; see the Appendix. the level sets of an arbitrary

quadratic polynomials for positive

test

elementary

functions

definite

(quadratic

definiteness

in two

forms)

of such

func-

tions:)

A homogeneous quadratic function V(XI, X2) c are real numbers, is positive + cx\037, where a, b, and - b 2 > o.) only if a > 0 and ac 9.11.

Lemma 2bxIX2

and

= definite

aXI + if

of the on the coefficients; conditions will prove the necessity that V is positive from similar follows reasoning. Suppose sufficiency If X2 =1= 0 is we must have a o. if 0 definite. Since V(XI, > > Xl =1= 0, 0) zeros Xl of be no real for all and there can then 0 > Xl fixed, V(XI, X2)

We

Proof.

the

V(XI, be

X2).

Thus

negative.

0

Now, we

the x =

the

would

level

sets

f(x),

then)

of

discriminant

like

to

4(b

2

-

ac) of

this quadratic function must

how the solutions of x = f(x) cross definite function V. If x( t) is a solution of

determine

a positive

8V

V(x(t)) = a Xl

(x(t))

Xl(t)

+

8V a X2)

(x(t)) X2(t).

(9.16))))

280

9:

Chapter

Near

Equilibria)

xO)

= k)

V(x)

= k)

V(x)

V(x)

V(XO)=O)

V(XO) 0, the orbit is crossing the level curve are shown in Figure 9.6. With these These three observations, possibilities is quite plausible: the following basic theorem of Liapunov

Theorem and V be

9.12.

(i) IfV(x) (ii) If V (x) (iii)

IfV(x)

Let x = 0 be an equilibrium point C 1 function on a neighborhood

(Liapunov)

definite

a positive

< 0

for


0

x

E U

x E

for x E

- {O}, then

U U

-

{O},

then

{O}, then

0

= f(x)

ofi. U

of

O.

is stable.

0 is

asymptotically

0 is

unstable.)

stable.

of the geometric remarks we will give a formal above, proof of the be small so that the Let \342\202\254 0 > neighborhood only. sufficiently in U. Let m be is contained origin consisting of the points with Ilxll < \342\202\254 = of this the minimum value of V on the boundary \342\202\254 neighborhood. Ilxll = \342\202\254 is closed definite and the set Ilxll and bounded, Since V is positive that such we have m > O. Now choose a 8 with 0 < 8 < \342\202\254 V(x) < m for = O. < is with 8. a 8 exists because V continuous Such always V(O) IIxll = xO satisfies of i. = f(x) with If Ilxoll < 8, then the solution x(t) x(O) < V(xO) for that for t > 0 since V(x(t)) < 0 implies V(x(t)) Ilx(t)11 < \342\202\254 t > O. This proves the stability of the equilibrium point at the origin. 0, there is a 8 > largest

0

eigenvalue of B, V(x)

+

2x

that

such

T

Bg(x).

Ilg(x)1I < mllxll if

then)

< -(1

-

T 2{3m)x x.)))

IIxll


0 for all x EOn U, then the origin is an unstable equilibrium point. (ii) V(x)

for

x on

all

of

0 inside

U;

illustrated in Figure 9.8 a typical situation dethe property (i), there are points in 0, hence that are arbitrarily close to the origin. From (ii) and (iii), no orbit of these from of 0 in U. points in 0 can cross the boundary starting anyone also from such orbits must leave the U Thus, neighborhood through (iii), 0. Consequently, the origin is an unstable equilibrium

X2

x E 0, and V(x) V along the solutions

the

and

x\037/2

-Xl

open

-1,

eigenof

none

region)

}.)

on the

== 0

the

Therefore, let us

is applicable.

X2) : Xl >

{ (Xl,

V(x) > 0 for of derivative

the origin. Notice that since at the origin are 0 and

sections

two

previous

function

the

consider

field

vector

linearized

theorems

the

at

point

equilibrium

285)

Functions

Liapunov

Next,

boundary. differential

of the

we

equations

above:)

It

is not

estimate)

.

X2) >

V(XI,

0 is small. By

c >

a quadratic

in

form

0

xI,

4 -

X\037) +

2

(1 +

x\037.

the

right-hand

it is

easy to

and thus

X2,)

side of

V(x)

E 0.

x

this inequality as

> 0 for x in a Now, the conditions

that

see

for

the

2

+

x I

c)IX2I

in particular,

and,

are satisfied

9.17

Theorem

==

Xl

viewing

and

IX21

of x

neighborhood of

-

X2(X\037

a sufficiently

in

where

-

xi

obvious that we are in the realm of Cetaev's theorem. small neighborhood U of the origin, we have the

immediately

However,

==

X2)

V(XI,

is unstable.

origin

0) .-.(/.0)

Exercises)

9.11.

Graphics: Computer

V(Xl,

X2)

==

V(Xl,

X2)

==

itive

definite.)

X\037

9.14. No

linear

determining cause the quadratic

(a) Xl

Use an

-

show

CX2

== ==

X\037+

+

(1

capabil-

2x\037; \037X\037;

\037X\037

-

COSXl).

where these

origin

a

are

functions

of odd

polynomial

pos-

degree can-

> 0,

the

-X\037 +

==

-X\037 + 2x\037,

(c) Xl

==

-

XlX\037,

X\037,

X2 == -2X\037X2 X2 == -2XlX\037;

consult

-

types x\037;

X2 == XlX\037 + 2X\037X2 + X\037.)))

system

of the

Section 8.3. is of no

linearization

below,

of the

\037 +00.

equilibrium point at is the zero matrix.

Jacobian matrix at the origin functions, determine the stability

==

solution

every

origin as t function;

quadratic

For the three systems the stability type of the

of Xl.)

power

for c

that,

part:

X\037

XlX2 + + \037X\037

homogeneous

approaches

appropriate

Xl

(b)

the

and factor out

9.12 to

== X2, X2 == -Xl

Suggestion:

of

x\037+

X2)

X2)

V(Xl,

X\037;

Show that a

== aXl,

X2

Use Theorem Xl

==

V(Xl,

of

graphs

definite.

positive

Hint: Take

X2)

V(Xl,

2)2;

neighborhoods

Odd polynomials: be

of a

4x\037;

a very

a standard

locate

+ X+ +

Find

not 9.13.

(Xl

X2) == X\037 the largest

V(Xl,

9.12.

x\037 +

useful tool for plotting package with 3-D graphics few functions, for example, is

graphics

functions. You should ities and plot the graphs

the However,

origin:

in

help

origin

beusing

286

Near

9:

Chapter

9.15. After

Equilibria)

the

Consider

Dirichlet:

and

Lagrange

Xl = X2,

=

X2

conservative

system)

-g(Xl),)

l

say, C . isolated minimum point Xl of the potential function xl du to a stable point equilibrium corresponds (Xl, 0) of the g(u) fo

where the

(a)

function

9 is,

each

that

Show

system. See

Help:

(b) Give

an

function 9 such that Xl is not a minimum and yet (Xl, 0) is a stable equilibrium point.

of a

function

potential

Hint: Try a 9.16.

14.

Chapter example

V(Xl,

X2) =

original

the

9.17. Consider

Can

X2).

-X\037V(Xl,

the

function.

nonanalytic

Suppose that you have the function whole (Xl, x2)-plane and that relative of the

of

differential

equation?

defined V(Xl, X2) = x\037e-Xl to some planar differential

you conclude anything about If not, what is the trouble?

the

on the equation solutions

of equations)

system

Xl =

X2

-

Xl!(Xl,

X2)

- x2f(xl,

X2 = -Xl

X2),)

Notice that the origin is an equilibrium real-valued C l function. function, point independent of the specific form of !. U sing a quadratic of the origin, then show that if f(xl, neighborhood X2) > 0 in some open the origin is asymptotically stable. What is the stability type of the origin if ! (Xl, X2) < 0 in a neighborhood of the origin?

is a

where!

9.18.

origin is an unstable

the

that

Show

.

= Xl3

Xl

.23 = -X2

+

X2

. V(Xl,

in a 9.19.

small

Actual determine of

the

6 X2) = Xl

+

>

-

x\037

basin of attraction: the largest

-

=

-

(1 +

the

= -Xl

X2 = =

1/9.)))

xt /4

x\037/2 2

+

Xl

-

show) X2)

> 0)

the

the basin 2 X2

-2X2 +

-

and

\037x\037

function

+

Xl')

+ x2(1

Xl)

origin, except

Using

.

x\037/4

X2)

+ XlX2

ellipse contained in

system)

+

X2

3 X2 X l

of the

neighborhood

xI/2

+

\037IX211xl13

Xl

Answer:

XlX2

the function V (Xl,

Consider

Hint:

for the system)

point

equilibrium

3x\037.)

origin

V(Xl,

itself. X2)

=

of attraction

xI/2 + of the

x\037/4,

origin

9.4.

to convince yourself that the than this is ellipse. However, origin larger there are other equilibrium because points.)

Experiment tion of the plane

that

Suppose

: IR \037 IR,

1/;

a

1---+

1/;(0'), ds

> 0 if a -# 0, and satisfying 1/;(0) = 0, 0'1/;(0') J: 1/;(s) with kp > 0, show constants For k, c, and p positive +00. of the indirect control problem)

-kx -

x =

=

\037,

\037

a = ex

1/;(0'),)

01

is a

entire

function

as

\037 +00

that

-

of attrac-

not the

it is

287)

Principle

basin

actual

numerically

control:

Indirect

9.20.

lnvariance

An

\037

10'1

solution

every

p\037,)

zero as t \037 +00. above control\" comes from the fact that in the system label \"indirect of the state variable the control variable \037is not given directly as a function another differential equation. In it is determined indirectly using X; instead, indirect control turns out to be very efficient; on related certain situations, matters, see, for example, Lefschetz [1965].) approaches

The

Invariance

An

9.4.

In this

section,

of the

discussion

detailed

exposition

in a

setting

the

case of

to

specialize

of Liapunov functions with a more stable of an asymptotically

our study

basin

In preparation

point.

equilibrium

Principle)

continue

we

is

that

attraction

of

for later chapters, we will commence for general limit sets and

equilibrium

points.)

U of JR2 is said to be spectively, negatively invariant] under the flow