Chen G., Huang Y. Chaotic Maps. Dynamics, Fractals, And Rapid Fluctuations (Morgan, 2011)(ISBN 159829914X)(O)(243s)_PNc

8/12/2019 Chen G., Huang Y. Chaotic Maps. Dynamics, Fractals, And Rapid Fluctuations (Morgan, 2011)(ISBN 159829914X)(O

1/243

CM&

Morgan Claypool Publishers&

SYNTHESISLECTURESONMATHEMATICS AND STATISTICS

Steven G. Krantz, Series Editor

Chaotic MapsDynamics, Fractals, andRapid Fluctuations

Goong Chen

Yu Huang


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Synthesis Lectures onMathematics and Statistics

EditorSteven G. Krantz, Washington University, St. Louis

Chaotic Maps: Dynamics, Fractals, and Rapid Fluctuations

Goong Chen and Yu Huang2011

Matrices in Engineering ProblemsMarvin J. Tobias2011

The Integral: A Crux for AnalysisSteven G. Krantz2011

Statistics is Easy! Second EditionDennis Shasha and Manda Wilson

2010Lectures on Financial Mathematics: Discrete Asset PricingGreg Anderson and Alec N. Kercheval2010

Jordan Canonical Form: Theory and PracticeSteven H. Weintraub2009

The Geometry of Walker ManifoldsMiguel Brozos-Vzquez, Eduardo Garca-Ro, Peter Gilkey, Stana Nikcevic, and RmonVzquez-Lorenzo

2009An Introduction to Multivariable MathematicsLeon Simon2008


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Jordan Canonical Form: Application to Differential EquationsSteven H. Weintraub2008

Statistics is Easy!Dennis Shasha and Manda Wilson2008

A Gyrovector Space Approach to Hyperbolic GeometryAbraham Albert Ungar2008


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Copyright 2011 by Morgan & Claypool

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in

any form or by any meanselectronic, mechanical, photocopy, recording, or any other except for brief quotations in

printed reviews, without the prior permission of the publisher.

Chaotic Maps: Dynamics, Fractals, and Rapid FluctuationsGoong Chen and Yu Huang

www.morganclaypool.com

ISBN: 9781598299144 paperback

ISBN: 9781598299151 ebook

DOI 10.2200/S00373ED1V01Y201107MAS011

A Publication in the Morgan & Claypool Publishers series

SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS

Lecture #11

Series Editor: Steven G. Krantz,Washington University, St. Louis

Series ISSN

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Chaotic MapsDynamics, Fractals, and Rapid Fluctuations

Goong ChenTexas A&M University

Yu HuangSun Yat-Sen University

SYNTHESIS LECTURES ON MATHEMATICS AND STATISTICS #11

CM

& cLaypoolMor gan publishe rs&


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ABSTRACTThis book consists of lecture notes for a semester-long introductory graduate course on dynamicalsystems and chaos taught by the authors at Texas A&M University and Zhongshan University,China. There are ten chapters in the main body of the book, covering an elementary theory ofchaotic maps in finite-dimensional spaces. The topics include one-dimensional dynamical systems(interval maps),bifurcations, general topological, symbolic dynamical systems, fractals and a class ofinfinite-dimensional dynamical systems which are induced by interval maps, plus rapid fluctuationsof chaotic maps as a new viewpoint developed by the authors in recent years.Two appendices are alsoprovided in order to ease the transitions for the readership from discrete-time dynamical systems tocontinuous-time dynamical systems, governed by ordinary and partial differential equations.

KEYWORDSchaos, interval maps, periodicity, sensitive dependence, stability, Sharkovskis theorem,bifurcations, homoclinicity, symbolic dynamics, smale horseshoe, total variations, rapidfluctuations, fractals, wave equation


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vii

ContentsPreface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xi

1 Simple Interval Maps andTheir Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Inverse and Implicit Function Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Visualizing from the Graphics of Iterations of the Quadratic Map . . . . . . . . . . . . 11

Notes for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Total Variations of Iterates of Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1 The Use of Total Variations as a Measure of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 21


3 Ordering among Periods:The Sharkovski Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 29Notes for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 BifurcationTheorems for Maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 The Period-Doubling Bifurcation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Saddle-Node Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 The Pitchfork Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


5 Homoclinicity. Lyapunoff Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1 Homoclinic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Lyapunoff Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


6 Symbolic Dynamics,Conjugacy and Shift Invariant Sets. . . . . . . . . . . . . . . . . . . . . 69

6.1 The Itinerary of an Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.2 Properties of the shift map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.3 Symbolic Dynamical Systems

kand+

k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 The Dynamics of(+

k, +)and Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81


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6.5 Topological Conjugacy and Semiconjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.6 Shift Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.7 Construction of Shift Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.8 Snap-back Repeller as a Shift Invariant Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Notes for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7 The Smale Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.1 The Standard Smale Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 The General Horseshoe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116


8 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.1 Examples of Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.2 Hausdorff Dimension and the Hausdorff Measure . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.3 Iterated Function Systems (IFS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130


9 Rapid Fluctuations of Chaotic Maps onRN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1419.1 Total Variation for Vector-Value Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

9.2 Rapid Fluctuations of Maps onRN

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1459.3 Rapid Fluctuations of Systems with Quasi-shift Invariant Sets . . . . . . . . . . . . . . 147

9.4 Rapid Fluctuations of Systems Containing Topological Horseshoes . . . . . . . . . . 149

9.5 Examples of Applications of Rapid Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 152


10 Infinite-dimensional Systems Induced by Continuous-Time DifferenceEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

10.1 I3DS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

10.2 Rates of Growth of Total Variations of Iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

10.3 Properties of the SetB(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

10.4 Properties of the SetU (f ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

10.5 Properties of the SetE(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Notes for Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177


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ix

A Introduction to Continuous-Time Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 179

A.1 The Local Behavior of 2-Dimensional Nonlinear Systems . . . . . . . . . . . . . . . . . . 179

A.2 Index for Two-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

A.3 The Poincar Map for a Periodic Orbit in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

B Chaotic Vibration of the Wave Equation due to Energy Pumping and van derPol Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

B.1 The Mathematical Model and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

B.2 Chaotic Vibration of the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Authors Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225


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PrefaceThe understanding and analysis of chaotic systems are considered as one of the most importantadvances of the 20th Century. Such systems behave contrary to the ordinary belief that the universeis always orderly and predicable as a grand ensemble modelizable by differential equations.The greatmathematician and astronomer Pierre-Simon Laplace (1749-1827) once said:

We may regard the present state of the universe as the effect of its past and the cause ofits future. An intellect which at a certain moment would know all forces that set nature

in motion, and all positions of all items of which nature is composed, if this intellect werealso vast enough to submit these data to analysis, it would embrace in a single formulathe movements of the greatest bodies of the universe and those of the tiniest atom; forsuch an intellect nothing would be uncertain and the future just like the past would bepresent before its eyes. (Laplace,A Philosophical Essay on Probabilities[47].)

Laplace had a conviction that, knowing all the governing differential equations and the initialconditions, we can predict everything in the universe in a deterministic way. But we now know thatLaplace has underestimated the complexities of the equations of motion. The truth is that rathersimple systems of ordinary differential equations can have behaviors that are extremely sensitiveto initial conditions as well as manifesting randomness. In addition, in quantum mechanics, eventhough the governing equation, the Schrdinger equation, is deterministic, the outcomes frommeasurements are probabilistic.

The term chaos, literally, means confusion, disarray, disorder, disorganization. turbulence,turmoil, etc. It appears to be the antithesis of beauty, elegance, harmony, order, organization, purity,and symmetry that most of us are all indoctrinated to believe that things should rightfully be.And, as such, chaos seems inherently to defy an organized description and a systematic study fora long time. Henri Poincar is most often credited as the founder of modern dynamical systemsand the discoverer of chaotic phenomena. In his study of the three-body problem during the 1880s,he found that celestial bodies can have orbits which are nonperiodic, and yet for any choices ofperiod of motion, that period will not be steadily increasing nor approaching a fixed value. Poincarsinterests have stimulated the development ofergodic theory, studied and developed by prominentmathematicians G.D.Birkhoff,A.N.Kolmogorov, M.L.Cartwright, J.E.Littlewood, S. Smale,etc.,

mainly from the nonlinear differential equations point of view.With the increasing availability of electronic computers during the 1960s, scientists and engi-

neers could begin to play with them and in so doing have discovered phenomena never known before.Two major discoveries were made during the early 1960s: theLorenz Attractor, by Edward Lorenz inhis study of weather prediction, andfractals, by Benot Mandelbrot in the study of fluctuating cotton


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xii PREFACE

prices. These discoveries have deeply revolutionized the thinking of engineers, mathematicians, and

scientists in problem solving and the understanding of nature and shaped the future directions inthe research and development of nonlinear science.

The actual coinage of chaos for the field is due to a 1975 paper by T.Y.Li and J.A.Yorke [49]entitled Period Three Implies Chaos. It is a perfect, captivating phrase for a study ready to take off,

with enthusiastic participants from all walks of engineering and natural and social sciences. Morerigorously speaking, a system is said to be chaoticif

(1) it has sensitive dependence on initial conditions;

(2) it must be topologically mixing; and

(3) the periodic orbits are dense.

Several other similar, but non-equivalent, definitions are possible and are used by different groups.Today,nonlinear scienceis a highly active established discipline (and interdiscipline), where

bifurcations, chaos, pattern formations, self-organizations, self-regulations, stability and instability,fractal structures, universality, synchronization, and peculiar nonlinear dynamical phenomena aresome of the most intensively studied topics.

The topics of dynamical systems and chaos have now become a standard course in both theundergraduate and graduate mathematics curriculum of most major universities in the world. Thisbook is developed from the lecture notes on dynamical systems and chaos the two authors taught atthe Mathematics Departments of Texas A&M University and Zhongshan (SunYat-Sen) Universityin Guangzhou, China during 1995-2011.

The materials in the notes are intended for a semester-long introductory course. The main

objective is to familiarize the students with the theory and techniques for (discrete-time) mapsfrom mainly an analysis viewpoint, aiming eventually to also provide a stepping stone for nonlinearsystems governed by ODEs and PDEs. The book is divided into ten chapters and two appendices.

They cover the following major themes:

(I) Interval maps: Their basic properties (Chapter 1), Sharkovskis Theorem on periodicities(Chapter 3), bifurcations (Chapter 4), and homoclinicity (Chapter 5).

(II) General dynamical systems and Smale Horseshoe: The 2- and k -symbol dynamics, topo-logical conjugacy and shift invariant sets (Chapter 6), and the Smale Horseshoe (Chapter7).

(III) Rapid fluctuations andfractals: Total variations and heuristics (Chapter 2), fractals (Chapter8),and rapidfluctuations of multi-dimensional maps and infinite-dimensional maps (Chapters9 and 10).

AppendixAis provided in order to show some basic qualitative behaviors of higher-dimensionaldifferential equation systems, and how to study continuous-time dynamical systems, which are often


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PREFACE xiii

described by nonlinear ordinary differential equations, using itsPoincar section, which is amap.This

would, hopefully, give the interested reader some head start toward the study of continuous-timedynamical systems.

AppendixBoffers an example of a concrete case of an infinite-dimensional system describedby the one-dimensional wave equation with a van der Pol type nonlinear boundary condition, andshows how to use interval maps and rapid fluctuations to understand and prove chaos.

For these three major themes (I)(III) above, much of the contents in (I) and (II) are ratherstandard. But the majority of the materials in Theme (III) is taken from the research done bythe two authors and our collaborators during the recent years. This viewpoint of regarding chaos asexponential growth of total variations on a strange attractorof some fractional Hausdorff dimensionsis actually mostly stimulated by our research on the chaotic vibrationof thewave equation introducedin AppendixB.

There are already a good number of books and monographs on dynamical systems and chaoson the market. In developing our own instructional materials, we have referenced and utilized themextensively and benefited immensely. We mention, in particular, the excellent books by Afraimovichand Hsu [2], Devaney[20], Guckenheimer and Holmes [30], Meyer and Hall [54], Robinson [58],and Wiggins [69]. In addition, we have also been blessed tremendously from two Chinese sources:

Wen [67]andZhou[75].To these book authors, and in addition,our past collaborators and studentswho helped us either directly or indirectly in many ways, we express our sincerest thanks.

Professor Steven G. Krantz, editor of the book series, and Mr. Joel Claypool, book publisher,constantly pushedus,tolerated therepeated long delays, butkindly expeditedthepublication process.

We are truly indebted.The writing of this book was supported in part by the Texas Norman Hackman Advanced Re-

search Program Grant #010366-0149-2009 from the Texas Higher Education Coordinating Board,Qatar National Research Fund (QNRF) National Priority Research Program Grants #NPRP09-462-1-074 and #NPRP4-1162-1-181, and the Chinese National Natural Science FoundationGrants #10771222 and 11071263.

Goong Chen and Yu HuangJuly 2011


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1

C H A P T E R 1

Simple Interval Maps and TheirIterations

The discovery of many nonlinear phenomena and their study by systematic methods are a majorbreakthrough in science and mathematics of the 20th Century, leading to the research and devel-opment ofnonlinear science, which is at the forefront of science and technology of the 21st Century.

Chaos is an extreme form of nonlinear dynamical phenomena. But what exactly ischaos? This is themain focus of this book.

Mathematical definitions of chaos can be given in many different ways. Though we will givethe first of such definitions at the end of Chapter 2 (inDef. 2.7),during much of the first few chaptersthe term chaos (or its adjective chaotic) should be interpreted in a rather liberal and intuitivesense that it stands for some irregular behaviors or phenomena. This vagueness should automaticallytake care of itself once more rigorous definitions are given.

1.1 INTRODUCTION

We begin by considering some population models. A simple one is the Malthusian law of linear

population growth: x0 > 0 is given;xn+1=xn; n=0, 1, 2, . . . (1.1)

where

xn=the population size of certain biological species at time n,and >0is a constant. For example, =1.03if

0.03=3%=net birth rate=birth ratedeath rate.

The solution of (1.1) is

xn=

nx0, n=

1, 2, . . . .

Therefore, xn , if >1, asn ,xn=x0, n=1, 2, . . . , if=1,xn0, if


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2 1. SIMPLE INTERVAL MAPS ANDTHEIR ITERATIONS

Thus, the long-term,orasymptotic behavior, of the system(1.1)is completely answered by (1.2).The

model(1.1)from the population dynamics point of view is quite naive. An improved model of (1.1)is the following:

xn+1=xn ax 2n,x0 >0 is given,

a >0, n=0, 1, 2, . . . (1.3)

where the term ax 2n modelsconflicts(suchas competitionforthesame resources) between membersof the species. It has a negative effect on population growth. Equation (1.3)is called themodified

Malthusian lawfor population growth.For a non-linear system with a single power law non-linearity, we can always scale out the

coefficient associated with the non-linear term.Let xn=kynand substitute it into(1.3).We obtain:

kyn+

1

=(kyn)

a(kyn)

2

yn+1=yn aky 2n.

Setk=/a . We have:yn+1=yn y2n= yn(1 yn).

Renameyn=xn. We obtain

xn+1=xn(1 xn)f (, xn), (1.4)

wheref(,x)=f(x)=x (1 x). (1.5)

The map fis called thequadratic maporlogistic map. It played a very important role in the devel-opment of chaos theory due to the study of the British biologist, Robert May, who noted (1975)that aschanges, the system does not attain simple steady states as those in (1.2). One of our maininterests here is to study theasymptotic behaviorof the iterates of (1.4) asn . Iterations of thetype xn+1=f (xn)happen very often elsewhere in applications, too. We look at another examplebelow.

Example 1.1 Newtons algorithm for finding the zero of a given functiong(x).

Newtons algorithm provides a useful way for approximating solutions of an equation g(x)=0iteratively. Start from an initial pointx0, we computex1, x2, . . ., as follows. At each point xn, drawa tangent line to the curvey=g(x)passing through (xn,g(xn)):

y g(xn)=g (xn)(x xn).

This line intersects thex-axis atx=xn+1:

0 g(xn)=g (xn)(xn+1 xn).


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1.1. INTRODUCTION 3

x

Zeros x,x

x

xn+1xn+2 xn

y= g(x)

x

Figure 1.1: Newtons algorithm.

So

xn+1=xn g(xn)

g(xn)f (xn),

where

f(x)x g(x)g(x)

.

The above iterations can encounter difficulty, for example, when:

(1) Atx, whereg(x)=0, we also have g (x)=0;

(2) The iteratesxnconverge to a different (undesirable solution)xinstead ofx;

(3) The iterates xj jump between two values xn and xn+1, such as what Fig.1.2shows in thefollowing.

If any of the above happens, we have:

limn

xn= x, for the desired solutionx.

From now on, for any real-valued functionf, we will use fn to denote then-th iterate offdefined by

fn(x)=f (f (f (f n-times

(x))) )=f f f f n-times

(x),


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x

xn

xn+1

y

x

Figure 1.2: Newtons algorithm becomes stagnant atxnand xn+1.

if eachfj(x)lies in the domain of definition off forj= 1, 2, . . . , n 1.

Exercise 1.2 Consider the iteration of the quadratic map:

xn+1=f(xn); f(x)=x (1 x),x0I [0, 1].

(1) Choose=3.2, 3.5, 3.55, 3.58, 3.65, 3.84, and 3.94. For each given, plot the graphs:

y=f(x),y=f2(x),y=f3(x),y=f4 (x), y=f5(x),y=f400 (x), x I ,

where

f2(x)=f(f(x)); f3(x)=f(f(f(x))), etc.

(2) Letbegin from=2.9and increaseto=4with increment=0.01,withas the

horizontal axis. For each, choose:

x0= k

100; k=1, 2, 3, . . . , 99.

Plotf400 (x0)(i.e., a dot) for these values ofx0on the vertical axis.


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1.1. INTRODUCTION 5

Example 1.3 The quadratic map fas defined in Exercise1.2and shown in Fig. 1.3is an example

of aunimodal map. A mapf: I [a, b] Iis said to be unimodal if it satisfiesf(a)=f (b)=a,

andfhas a unique critical point c : a < c < b. The quadratic mapfis very representative of thedynamical behavior of unimodal maps.

Let fbe a continuous function such that f: I Ion a closed interval I.A pointxis said to be afixed pointof a mapy=f (x)if

x=f (x). (1.6)The set of all fixed points offis denoted as Fix(f ). A pointx is said to be aperiodic pointwithprime period k, if

x=fk(x), (1.7)andkis the smallest positive integer to satisfy (1.7).The set of all periodic points of prime period koff is denoted as Perk (f ), and that of all periodic points off is denoted as Per(f ). In the analysisof the iterationsxn+1=f (xn),fixed points and periodic points play a critical role.

Look at the quadratic mapf(x)in Fig.1.3.

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fixed point, 1

x(1x)

y

y = x

Figure 1.3: Graph of the Quadratic Mapy=f(x), =2.7. Its maximum always occurs at x=1/2.It has a (trivial) fixed point at x=0, and another fixed point at x= 1

.

The fixed pointx= 1 can beattractingorrepelling, as Fig.1.4(a) and (b) have shown.Definition 1.4 Letx be a periodic point of prime period p of a differentiable real-valued mapf: fp(x)= x. We say thatxis attracting (resp., repelling) if

|(fp)(x)|< 1 (resp., |(fp)(x)|> 1).


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0 0.2 0.4 0.6 0.8 10

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(a)

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0.1

0.2

0.3

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0.7

0.8

0.9

1

(b)

Figure 1.4: (a) The fixed point has slope f(x) such that|f(x)|< 1. The iterates are attracted to thefixed point.(b) The fixed point has slopef(x)such that|f(x)|> 1. The iterates are moving away fromx.

A periodic pointxof prime periodnis said to behyperbolicif|(fp)(x)| =1, i.e., xmust be eitherattracting or repelling.

We present a few fundamental theorems.

Theorem 1.5 (Brouwers Fixed PointTheorem) Let I= [a, b],andlet fbe a continuous functiononI

such that either (i) f ( I )

Ior (ii) f( I )

I .

Thenfhas at least one fixed point on I.

Proof. Consider (i) first:Define:g(x)=x f(x). Because f( I )I, i.e.,f ([a, b]) [a, b], so f(a) [a, b], and

f(b) [a, b], thusaf(a)b, af (b)b.Then

g(a)=a f(a)0,g(b)=b f(b)0.

If equality does not hold in either of the two relations above, then

g(a) 0.

By the Intermediate Value Theorem, we get

g(c)=0 for some c : a < c < b.


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1.1. INTRODUCTION 7

Thus, cis a fixed point off. Next consider (ii):

f( I )I , [a, b] f ([a, b]).

Therefore, there existx1, x2 [a, b]such that

f (x1)a < bf (x2).

Again, defineg(x)=x f(x).Then

g(x1)=x1 f (x1)x1 a0,g(x2)=x2 f (x2)x2 b0.

Therefore, there exists a pointxin either[x1, x2]or[x2, x1], such thatg(x)=0.Thus,xis a fixedpoint off.

Theorem 1.6 Let f: I Ibe continuous where I= [a, b], such that f is also continuous,satisfying:

|f(x)|< 1 on I.

Thenfhas a unique fixed point on I.

Proof. The existence of a fixed point has been proved in Theorem 1.5,so we need only proveuniqueness. Suppose bothx0and y0are fixed points off:

x0=f (x0),y0=f (y0).

Thenf (y0) f (x0)

y0 x0= y0 x0

y0 x0=1=f(c)

by the Mean Value Theorem, for somec : x0 < c < y0. But

|f(c)|< 1.

This is a contradiction.


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1.2 THE INVERSE AND IMPLICIT FUNCTION THEOREMSFrom now on, we denote vectors and vector-valued functions by bold letters.

We state without proof two theorems which will be useful in future discussions.

Theorem 1.7 (The Inverse Function Theorem) Let U and Vbe two open sets in RN andfff: U V isC r for somer 1. Assume that

(i) xxx0 U , yyy0 V, and fff (xxx0)= yyy0;

(ii)fff (xxx)|xxx=xxx0is nonsingular, where

fff (xxx)=

f1(xxx)

x1 f1(xxx)

xNf2(xxx)

x1 f2(xxx)

xN...

...fN(xxx)

x1 fN(xxx)

xN

.

Then there exists an open neighborhoodN (xxx0)Uofxxx0 and an open neighborhoodN (yyy0)Vofyyy0 and aC r -map ggg:

ggg : N (yyy0)N (xxx0),such that

fff (ggg(yyy))= yyy,i.e., gggis a local inverse offff .

Theorem 1.8 (The Implicit Function Theorem) Let

f1(x1, . . . , xm, y1, . . . , yn)=0,f2(x1, . . . , xm, y1, . . . , yn)=0,

......

fn

(x1

, . . . , xm

, y1

, . . . , yn

)=

0

(1.8)

be satisfied for allxxx=(x1, . . . , xm)Uandyyy=(y1, . . . , yn)V, whereUandVare open setsin, respectively,Rm and RN, and

fi : U V R isC r , for somer 1, for all i=1, 2, . . . , n .


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1.2. THE INVERSE ANDIMPLICIT FUNCTION THEOREMS 9

Assume that for xxx0

=(x 0

1

, x0

2

, . . . , x0m)

Uand yyy0

=(y 0

1

, y0

2

, . . . , y0n)

V,

fi (xxx0, yyy0)=0 for i=1, 2, . . . , n ,

[y fi (xxx0, yyy0)] =

f1

y1 f1

yn...

...fn

y1 fn

yn

at x=x0x=x0x=x0

y=y0y=y0y=y0

is nonsingular.

Then there exist an open neighborhoodN (xxx0)Uofxxx0 and an open neighborhood N (yyy0)Vofyyy0, and aC r -map ggg : N (xxx0)N (yyy0), such that yyy0 = ggg(xxx0)and

fi (xxx, ggg(yyy))=0, xxxN (xxx0

), yyy N (yyy0

), i=1, 2, . . . , n ,i.e., locally, yyyis solvable in terms ofxxxbyyyy= ggg(xxx).

If we write equations in (1.8) as

FFF (xxx, yyy)=(f1(xxx, yyy),f2(xxx, yyy ) , . . . , f n(xxx, yyy))= 000,then taking the differential around xxx= xxx0 and yyy= yyy0, we have

xxxFFF dxxx+yyyFFF dyyy= 000,wheredxxx xxxxxx0 anddyyy yyyyyy0.Thus,

xxxFFF (xxxxxx0

) + yyyFFF (yyyyyy0

)= 000.An approximate solution ofyyyin terms ofxxxnear yyy= yyy0is

yyy yyy0 + [yyyFFF (xxx0, yyy0)]1xxxFFF (xxxxxx0).This explains intuitively why the invertibility ofyyyFFF (xxx0, yyy0)is useful.

The Implicit Function Theorem can be proved using the Inverse Function Theorem, but theproofs of both theorems can be found in most advanced calculus books so we omit them here.

Example 1.9 Consider the relation

f( x , y )=ax 2 + bx+ c + y=0. (1.9)

We havef

x=2ax+ b=0, if x= b

2a;

f

y=1=0.


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Thus, yis always solvable in terms ofx :

y= (ax2 + bx+ c).

Here, we actually see that if x=x 0 = b2a

, then x is not uniquely solvablein terms of y in aneighborhood ofx 0 = b

2abecause by the quadratic formula applied to(1.9), we have

x=b

b2 4a(c + y)2a

= b2a

b2 4a(c + y)2a

,

i.e., xis not unique.On the other hand,ifx0 = b

2a, then in a neighborhood ofx0, xis uniquely solvable in terms

ofy . For example, for a=b=c=1, withx 0 =2andy 0 = 7,

x0 =2= b2a

= 12

.

The (unique) solution ofx in terms ofy in a neighborhood ofx is thus

x=b +

b2 4a(c + y)2a

=1 +

1 4(1 + y)2

.

We discard the branch

x=b

b2 4a(c + y)2a

=1

1 4(1 + y)2

because it doesnt satisfy

x0 =2=1 +

1 4(1 7)2

=1 + 52

.

Exercise 1.10 Assume thata , b, c, d R, anda=0. Consider the relation

f( x , y )=ax 3 + bx 2 + cx+ d y=0, x, y R

(i) Discuss the local solvability of real solutions x for given y by using the implicit functiontheorem.

(ii) Under what conditions does the function

y=g(x)=ax 3 + bx2 + cx+ d

have a local inverse? A global inverse?


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1.3. VISUALIZING FROMTHE GRAPHICSOF ITERATIONS OFTHE QUADRATIC MAP 11

Figure 1.5: The orbit diagram off (, x)=f(x)=x (1 x).

1.3 VISUALIZING FROM THE GRAPHICS OF ITERATIONSOF THE QUADRATIC MAP

In the next few pages, we discuss the computer graphics from the previous Exercise1.2.

This type of graphics in Fig. 1.5 iscalledan orbit diagram. Note that thefirst perioddoubling happensat 0=3. Then the second and third happen, respectively, at 13.45and 23.542, and moreperiod doublings happen in a cascade. We have

1 02 1

3.45 33.542 3.45 =

0.45

0.0924.8913 . . . .

It has been found that forany period-doubling cascade,

limn

n n1n+1 n

=4.669202 . . . .

This number, auniversal constantdue to M. Feigenbaum, is called the Feigenbaum constant. Also,note that there is a window area near

=3.84in Fig.1.5.


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0 0.5 10

0.2

0.4

0.6

0.8

1

xaxis

f2

(x) f3

(x)

f4

(x) f5

(x)

0 0.5 10

0.2

0.4

0.6

0.8

1

xaxis0 0.5 1

0

0.2

0.4

0.6

0.8

1

xaxis

f(x)

0 0.5 10

0.2

0.4

0.6

0.8

1

xaxis

0 0.5 10

0.2

0.4

0.6

0.8

1

xaxis

Figure 1.6: The graphics of f(x), f2 (x), f3

(x), f4

(x) and f5

(x), where =3.2. Note that theintersections of the curves with the diagonal liney=x represent either a fixed point or a periodic point.

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1f(x) f

2(x) f

3(x)

f4

(x) f5

(x)

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

Figure 1.7: The graphics off(x), f2(x), f3

(x), f4

(x)andf5

(x), where=3.40.


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0 0.5 10

0.2

0.4

0.6

0.8

1

xaxis0 0.5 1

0

0.2

0.4

0.6

0.8

1

xaxis0 0.5 1

0

0.2

0.4

0.6

0.8

1

xaxis

0 0.5 10

0.2

0.4

0.6

0.8

1

x

axis

0 0.5 10

0.2

0.4

0.6

0.8

1

x

axis

f(x) f2

(x)

f4

(x) f5

(x)

f3

(x)

Figure 1.8: The graphics off(x),f2(x),f3

(x),f4

(x) and f5

(x) where=3.84. We see that thecurves have become more oscillatory (in comparison with those in Fig.1.7). They intersect with thediagonal line y=x at more points, implying that there are more periodic points. Note that f3(x)intersects withy=x at P1, P2, . . . , P 5and P6(in addition to the fixed pointx=0). Each point Pi ,i=1, 2, . . . , 6,hasperiod3.If a continuous map has period 3, then it has periodnfor anyn=1, 2, 3, 4, . . . .


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xaxis

f400

(x)

Figure1.9:The graph off400 (x),where =3.2. It looks like a step function.The two horizontal levelscorrespond to the period-2 bifurcation curves in Fig.1.5.Question: In thex -ranges close tox=0andx=1, how oscillatory is the curve?


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xaxis

f400

(x)

Figure 1.10: The graph off400 (x), where =3.5. It again looks like a step function, but with fourhorizontal levels.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xaxis

f400

(x)

Figure 1.11: The graph off400 (x), with=3.55. This curve actually haseighthorizontal levels.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xaxis

f400

(x)

Figure1.12: The graph off400 (x), =3.65.This value of is already in the chaotic regime.The curvehas exhibited highly oscillatory behavior.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xaxis

f400

(x)

Figure 1.13: The graph off400 (x),=3.84. Note that this value of corresponds to the windowarea in Fig.1.5.The curve is highly oscillatory, but it appears to take only three horizontal values.


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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

xaxis

f400

(x)

Figure 1.14: The graph off400 (x), with=3.93. This value ofis also in the chaotic regime.


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Visualization from these graphics, Figs.1.51.14, will provide inspirations for the study of

the oscillatory behaviors related to chaos in this book.

Exercise 1.11 Define

y=f(x)= sin x, x I= [0, 1].

(1) Vary [0, 1], plot the graphics off, f2, f3, . . . , f 10 .(2) Plot the orbit diagrams off(x).

(3) Describe what happens if >1.

Exercise 1.12 Pick your arbitrary favorite continuous function of the form

y=f (, x), x [0, 1],

such thatf (, )mapsIintoIfor the parameterlying within a certain range. Plot the graphicsof the various iterates off (, )as well as orbit diagrams off (, ).

NOTES FOR CHAPTER 1Aninterval mapis formed by the 1-step scalarequation of iterationxn+1=f (xn)for a continuousmap f. Thus, it constitutes the simplest model for iterations. For example, Newtons method for

finding roots of a nonlinear equation, and the time-marching of a 1-step explicit Euler finite-difference scheme for a first order scalar ordinary differential equation, can both result in an intervalmap. Interestingly, even for partial differential equations such as the nonlinear initial-boundary valueproblem of the wave equation in AppendixB, interval maps have found good applications.

Most of the textbooks on dynamical systems use the quadratic (or logistic) map (1.4) as astandard example to illustrate many peculiar, amazing behaviors of the iterates of the quadratic map.In fact, those iterates manifest strong chaotic phenomena which facilitates the understanding of

what chaos is for pedagogical purposes. The focus of the first five chapters of this book is almostexclusively on interval maps.

The books by Devaney [20] and Robinson[58] contain excellent treatments of interval maps.The monograph by Block and Coppel [7] contains a more detailed account and further references

about interval maps.


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21

C H A P T E R 2

Total Variations of Iterates ofMaps

2.1 THE USE OFTOTAL VARIATIONS AS A MEASURE OFCHAOS

Let f: I= [a, b] Rbe a given function; f is not necessarily continuous. ApartitionofI isdefined as

P={x0, x1, . . . , xn|xj I , forj= 0, 1, . . . , n; a=x0 < x1 < x2 < < xn =b},

which is an arbitrary finite collection of (ordered) points onI. Define

VI(f )=the total variation offonI

= supallP

ni=1

|f (xi ) f (xi1)| xi P

. (2.1)

Iff is continuous on I, and f has finitely many maxima and minima on I, such as indicated in

Fig.2.1below. Then it is easy to see that

VI(f )= |f (x1) f (x0)| + |f (x2) f (x1)| + + |f (xn) f (xn1)|,

where each interval[xi ,xi+1]is a maximal interval wherefis either increasing or decreasing.Let I1and I2be two closed intervals, and letfbe continuous such that

f (I1)I2.

We write the above asI1

fI2 or I1I2and say thatI1 f-coversI2.

Lemma 2.1 IfI1fI2, thenVI1 (f ) |I2| length ofI2.

Proof. This follows easily from the observation of Fig.2.2.


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22 2. TOTAL VARIATIONS OF ITERATESOF MAPS

x2 x3 x4 xn2x5 xn1

f

xnxn3x0 x1

Figure 2.1: A continuous function with finitely many maxima and minima atx0,x1, . . . ,xn.

I2

I1

Figure 2.2: IntervalI1 f-coversI2.

Lemma 2.2 Let J0, J1, . . . , J n1be bounded closed intervals such that they overlap at most atendpoints pairwise. Assume thatJ0J1J2 Jn1JnJ0holds. Then

(i) there exists a fixed point x0offn : fn(x0)=x0, such thatfk(x0)Jkfor k=0, . . . , n;

(ii) Further, assume that the loop J0

J1

Jn is not a repetition of a shortened

repetitive loopmwheremk=nfor some integerk >0. If the pointx0in (i) is in the interiorofJ0, thenx0has prime periodn.

Proof. Use mathematical induction.


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2.1. THEUSE OFTOTAL VARIATIONS AS A MEASURE OF CHAOS 23

Theorem 2.3 Let Ibe a closed interval and f:

I

Ibe continuous. Assume that fhas two

fixed points onIand a pair of period-2 points on I.Then

limn VI(f

n)= .

Proof. Let the two fixed points bex0and x1:

f (x0)=x0 and f (x1)=x1. (2.2)

Let the two period-2 points be p1and p2:

f (p1)=p2, f (p2)=p1. (2.3)

Then there are three possibilities:

(i) p1 < x0 < x1 < p2; (2.4)(ii) x0 < p1 < p2;(iii) p1 < p2 < x1.

We consider case (i) only. Cases (ii) and (iii) can be treated in a similar way.

p1 x0 x1 p2

I1

I3I2

Figure 2.3: The points and intervals corresponding to (2.2), (2.3) and (2.4).

From Fig.2.3,we see that we have

f (I1)I2 I3, i.e., I1I2 I3,f (I2)I2, i.e., I2I2,

f (I3)I1 I2, i.e., I3I1 I2.

(2.5)

Therefore, we have the covering diagramIt is easy to verify by mathematical induction that the following statement is true:

For eachn, I1containsn + 1subintervals{I(n)1,1 ; I(n)1,2 , . . . , I (n)1,n+1}such that


40/243


I

I

I

I

I

I

I

I

ffff

I

I

I

I

I

I

I

32 4

1 1 1

2

2

2

2

2

2

2

2

2

3 3

2

...

.

.

.

.

.

.

.

Figure 2.4: The covering of intervals according to (2.5).

I(n)1,j I1, I(n)1,j1 I

(n)1,j2

has empty interior ifj1=j2,fn(I

(n)1,j

)Ik for some k {1, 2, 3}.Therefore,

VI(fn)VI1 (fn)

n+1j=1

VI

(n)1,j

(fn)

(n + 1) min{|I1|, |I2|, |I3|} asn . (2.6)

Exercise 2.4 Prove that for the quadratic map

f(x)=x (1 x), xI= [0, 1],if >3,then f hastwofixedpointsandatleastapairofperiod-2points.Therefore,theassumptionsof Theorem2.3are satisfied and

limn VI(f

n )= for all : 3 < 0, (2.7)

i.e., the total variation offn grows exponentially withn.


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p p p pp p23 3121

Figure 2.5: Two period-3 orbits satisfyingf (p1)=p2, f ( p2)=p3, f ( p3)=p1.

Proof. To help our visualization, we draw the graphics of a period-3 orbit in Fig.2.5.We have two possibilities:

(i) p2 < p1 < p3, or (ii) p3 < p1 < p2. (2.8)

Here we treat only case (i). Define

I1= [p1, p2], I2= [p3, p1].

Thenf (I1)I1 I2, i.e., I1I1 I2,f (I2)I1, i.e., I2I1.

(2.9)

Thus, we have the covering diagram in Fig.2.6.

For each n, one can prove by mathematical induction that if the (n + 1)th column (after mappingbyfn) containsansubintervals ofI1or I2, then the following relation is satisfied: an+1=an+ an1, forn=2, 3, 4, . . . ,a1=2, a2=3. (2.10)

An exact solution to the recurrence relation (2.10) can be determined as follows. Assume that asolution ofan+1=an+ an1can be written in the form

ak= cx k , for k=1, 2, . . . . (2.11)

Then substituting(2.11) into the first equation of (2.10) gives

cx n+1 =cx n + cx n1,x2 x 1=0,x= 1

5

2.


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I1

1I

1

I1

ff3

f f2 4

I2

I1

I2

I1

I2

II

2

1

I2

I

I

2I

2

I

I2

I

1

I1

1I1

. . .

. . .

. . .

52 3 intervals8

Figure 2.6: The covering diagram satisfying (2.9).

Therefore, we write the solution of(2.10) as

an=c1

1 + 52

n+ c2

1 5

2

n, n=1, 2, . . . ,

a1=2=c11 + 5

2

+ c2

1 52

,

a2=3=c1

1 + 52

2+ c2

1 5

2

2,

and obtain

c1= 5 + 35

10, c2=

5 3510

.

It is easy to show that

an=c1 1 + 52

n + c2 1 52

nk0

1 + 5

2

n


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for some k0 >0, for all n=

1, 2, . . .. Therefore, using the same arguments as in the proof of

Theorem2.3,we have

VI(fn)VI1 (fn)k0

1 + 5

2

n min{|I1|, |I2|} ken, forn=1, 2, . . .

where

kk0 min{|I1|, |I2|} and ln

1 + 52

> 0. (2.12)

Using the same arguments as in the proof ofTheorem2.5, we can also establish the following.

Theorem 2.6 Let Ibe a bounded closed interval and f: I Ibe continuous. Assume thatI1, I2, . . . , I n are closed subintervals of Iwhich overlap at most at endpoints, and the coveringrelation

I1I2I3 InI1 Ij, for somej=1. (2.13)

Then for someK >0and >0,

VI(fn)K en , as n . (2.14)

Theorem2.6motivates us to give the following definition of chaos, the first one of suchdefinitions in this book.

Definition 2.7 Let f: I Ibe an interval map such that there exist K >0, >0 such that(2.14) holds. We say thatf ischaotic in the sense of exponential growth of total variations of iterates.

In Chapter9,such a mapfwill also be said to have rapid fluctuations of dimension 1.

Corollary 2.8 Let f: I I be an interval map satisfying(2.13)in the assumption of Theo-rem2.6.Thenf is chaotic in the sense of exponential growth of variations of iterates.


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NOTES FOR CHAPTER 2The total variation of a scalar-valued function on an interval provides a numerical measure of howstrong the oscillatory behavior that function has, when the interval is finite. This chapter is based onG. Chen, T. Huang and Y. Huang [17]. It shows that the total variations of iterates of a given mapcan be bounded, of polynomial growth, and of exponential growth. Only the case of exponentialgrowth of total variations of iterates is classified as chaos (while the case of polynomial growth isassociated with the existence of periodic points).This offers aglobal approachto the study of Chaoticmaps.

This chapter will pave the way for the study of chaotic behavior in terms of total variations inhigher and fractional dimensions in Chapters8and9.


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29

C H A P T E R 3

Ordering among Periods: TheSharkovskiTheorem

One of the most beautiful theorems in the theory of dynamical systems is the Sharkovski Theorem.An interval map may have many different periodic points with seemingly unrelated periodicities.

What is unexpected and, in fact, amazing is that those periodicities are actually oredered in a certain

way, called the Sharkovski ordering. The top chain of the ordering consists of all odd integers, withthe number 3 at the zenith, and the bottom chain of the ordering consists of all decreasing powersof 2, with the number 1 at the nadir.

Here we give the statement of the theorem and provide a sketch of ideas of the proof. Weintroduce the Sharkovski ordering on the set of all positive integers.The ordering is arranged as inFig.3.1.

3

5

7

9...

2n + 1

2n + 3

...

2 3

2 5

2 7

2 9...

2 (2n + 1)

2 (2n + 3)

...

22 3

22 5

22 7

22 9...

22 (2n + 1)

22 (2n + 3)

...

.

.

.2n 3

2n 5

2n 7

2n 9...

2n (2m + 1)

2n (2m + 3)

...

.

.

.

.

.

.

2m+1

2m

2

m

1

...

23

22

2

1.

Figure 3.1: The Sharkovski ordering.

Theorem 3.1 (Sharkovskis Theorem) Let Ibe a bounded closed interval and f: I Ibecontinuous. Letn kin Sharkovskis ordering. Iffhas a(prime)periodnorbit, thenfalso has a(prime)periodkorbit.


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30 3. ORDERINGAMONG PERIODS:THE SHARKOVSKI THEOREM

The following lemma is key in the proof of Sharkovskis Theorem.

Lemma 3.2 Let n be an odd integer. Let fhave a periodic point of prime period n. Then thereexists a periodic orbit{xj| j= 1, 2, . . . n;f (xj)=xj+1for j= 1, 2, . . . , n 1;f (xn)=x1}ofprime periodnsuch that either

xn


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31

However,In1= [xn, xn2], f (xn)=x1, f (xn2)=xn1,

and, thereforef (In1) [x1, xn1] =I1 I3 I5 In2. (3.6)

Example 3.3 In Lemma3.2,let n=7. Then (3.3)(3.6) above give us the following diagram inFig.3.3.

I

I

I

I

I

6I

1

2

34

5

Figure 3.3: Covering relations for intervals I1, I2, . . . , I 6 where n=7 in Lemma3.2.Note that I1covers bothI1and I2, whileI6covers all the odd-numbered intervalsI1, I3and I5.

For a general odd positive integer n, from(3.3)(3.6), we can construct the graph in Fig. 3.4,called theStefan cycle.

In2

In1

I1

I2

I3

I4

I5

...

. ..

Figure 3.4: The Stefan cycle corresponding to(3.1).


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32 3. ORDERINGAMONG PERIODS:THE SHARKOVSKI THEOREM

Proposition 3.4 Assume thatn is odd and k is a positive integer such that n

kin Sharkovskis

ordering. Iffhas a prime period n, thenfalso has a prime periodk .

Proof. There are two possibilities: (i) k is even and k < n; and (ii)k > nand k can be either evenor odd.

Consider Case (i) first. We use the loop

In1InkInk+1 In2In1. (3.7)

note:n khere is odd.Then by Lemma2.2in Chapter2,there exists anx0In1with periodk. The pointx0cannot bean endpoint because the endpoints have period n. Therefore, x0has prime periodk .

Next, we consider Case (ii). We use the following loop of lengthk:

I1I2 In1I1I1 I1 kn+1 I1 s

(3.8)

Thus, there exists anx0I1withfk(x0)=x0. Ifx0is an endpoint ofI1,thenx0has periodnand,therefore,kis divisible byn, andk2nn + 3. Eitherx0=x1or x0=x2is satisfied forx1andx2in (3.1). Thus,

(a) ifx0=x1, thenfn(x0)=x0and fn+2(x0)=fn+2(x1)=f2(x1)=x3;(b) ifx0=x2, thenfn(x0)=x0and fn+2(x0)=fn+2(x2)=f2(x2)=x4.

In either case above,fn

+2(x

0) /

I1. This violates our choice of

x0that it satisfies

x0I1 I2

f (x0)

I3

f2(x0)

In1 I1

fn1(x0 )

I1

fn(x0)

I1

fn+1(x0)

I1

fn+2(x0)

I1,

from (3.8) becausek nn + 3. For readers who are interested to see a simple, complete proof of the Sharkovski Theorem, we

recommend Du [23,24,25].

NOTES FOR CHAPTER 3A.N. Sharkovski (1936-) published his paper [62] in the Ukrainian Mathematical Journal in 1964.This paper was ahead of the time before iterations, chaos and nonlinear phenomena became fash-ionable. Also, the paper was written in Russian. Thus, Sharkovskis results went unnoticed for adecade.


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33

The American mathematicians Tien-Yien Li and James A. Yorke published a famous paper

entitled Period three implies chaos in the American Mathematical Monthly in 1975.They actuallyproved a special part of Sharkovskis result besides coining the term chaos. Li and Yorke attendeda conference in East Berlin where they met Sharkovski. Although they could not converse in acommon language, the meeting led to global recognition of Sharkovskis work.

Sharkovskis Theorem does not hold for multidimensional maps. For circle maps, rotation byone hundred twenty degrees is a map with period three. But it does not have any other periods.

Today, there exist several ways of proving SharkovskisTheorem: by Stefan [66],Block,Guck-enheimer, Misiurewicz and Young [8], Burkart [9],HoandMorris[33], Ciesielski and Pogoda [19],and Du [23,24], and others.


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35

C H A P T E R 4

Bifurcation Theorems for MapsBifurcation means branching. It is a major nonlinear phenomenon. Bifurcation happens when oneor several important system parameters change values in a transition process. After a bifurcation, thesystems behavior changes. For example, new equilibrium states emerge, with a different behavior,especially that related to stability.

4.1 THE PERIOD-DOUBLING BIFURCATION THEOREMPeriod doubling is an important chaos route to. We have seen from Fig.1.4that the (local) diagramlooks like what is shown in Fig.4.1.

C1

periodic-2 points

0

C2

fixed p oints

C3

C1

Figure 4.1: Period doubling and stability of bifurcated solutions.

In Fig.4.1,theC1branch of fixed points loses its stability at =0and bifurcates into C2 C3,which is a curve of period-2 points. We want to analyze this bifurcation. In performing the analysis,there are at least two difficulties involved:

(i) For the iterationxn+1=f(xn)=f (, xn), any period-2 pointxsatisfies

x=f2 (x)=f (, f (,x)). (4.1)

But ifxis a fixed point,xwill also satisfy(4.1). How do we pick out those period-2 pointsxwhich are not fixed points?

(ii) Assume that we can resolve (i) above. Can we also determine the stability (i.e., whetherattracting, or repelling) of the period-2 points?


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36 4. BIFURCATION THEOREMSFOR MAPS

These will be answered in the following theorem.

Theorem 4.1 (Period Doubling Bifurcation Theorem) Consider the mapf(, ) : I I where I is a closed interval and f is Cr for r 3. Let the curve C repre-sent a family of fixed points off (, ), whereC : x=x(), andx()satisfies

x()=f (, x()).Assume that at =0,

(i) f (, x)

x

=0x=x(0)x0

= 1, (4.2)

(ii) 2f(,x)

x +1

2

f(,x)

2f(,x)

x2 =0x=x0

==0, (4.3)

then there is period-doubling bifurcation at =0 and x=x0, i.e., there exists a curve ofthe parameter in terms ofx : =m(x)in a neighborhood of =0 and x=x0 such that0=m(x0)and

x=f2(x)|=m(x)=f (,f (,x))|=m(x). (4.4)Further, assume that

1

3!3f(,x)

x3 +

1

2! 2f(,x)

x2

2

=0x=x0

==0. (4.5)

Then the bifurcated period-2 points on the curve =m(x)are attracting if >0and repelling if


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4.1. THE PERIOD-DOUBLING BIFURCATIONTHEOREM 37

This gives us the curve C:

x

=x()of fixed points.

Next, we want to capture the bifurcated period-2 points near x=x0 and =0. Thesepoints satisfy(4.1). To simplify notation, let us define

g(, y)=f (, y+ x()) x(). (4.7)Then it is easy to check that

j

y jg(,y )

y=0

= j

xjf (,x)

x=x()

, for j= 1, 2, 3, . . . . (4.8)

This change of variable will give us plenty of convenience. We note thaty=y()0becomes thecurve of fixed points for the map g.Since g(,y)|y=0=g(, 0)=0, we have the Taylor expansion

g(, y)=a1()y+ a2()y2 + a3()y3 +O(|y|4).The period-2 points offnow satisfy

y=g 2(y)=g(,g(,y))=a1()[a1()y+ a2()y2 + a3()y3 +O(|y|4)]

+ a2()[a1()y+ a2()y2 + a3()y3 +O(|y|4)]2+ a3()[a1()y+ a2()y2 + a3()y3 +O(|y|4)]3 +O(|y|4)

y=a 21 y+ (a1a2+ a21 a2)y2 + (a1a3+ 2a1a22+ a31 a3)y3 +O(|y|4),where in the above, we have omitted the dependence ofa1, a2and a3on . Since y=y()=0corresponds to the fixed points off, we dont want them. Therefore, define

M(, y)

=

g2(y) y

y

if y

=0. (4.9)

This gives

M(, y)=(a 21 1) + (a1a2+ a21 a2)y+ (a1a3+ 2a1a22+ a31 a3)y2 +O(|y|3).The above functionM(, y)is obviouslyC 2 even fory=0. Thus, we can extend the definition ofM(,y)given in (4.9)by continuity even toy=0. Now, note that period-2 points are determinedby the equationM(, y)=0. We have

M(, y)

=0y=0

=2a1(0)a1(0)

=2

g(,y)

y =0y=0

2g(,y)

y =0y=0

=2

f (0, x0)

x

2f (0, x0)

x+

2f (0, x0)

x2 x (0)

= 2, (4.10)


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where we have utilized the fact that

x()=f (,x()),x()= f

+ f

xx(),

x()= f/1 f

x

,

x (0)= f (0, x0)/

1 f (0,x0)x

= 12

f (0, x0)

.

From (4.3) and(4.10), we thus have

M(, y)=0y=0

= 2=0. (4.11)

On the other hand,

yM(,y)

=0y=0

=a1(0)a2(0) + a1(0)2a2(0)

= a2(0) + a2(0)=0. (4.12)

From (4.11) and the Implicit Function Theorem, we thus conclude that near =0and y=0,there exists a curve

:

=m(y)such that

M(m(y), y)=0, (4.13)

i.e., =m(y)represents period-2 points of the mapf. We have

m(0)=0,

andm(0), m(0)may be computed from (4.13):

0=

M

m(y) + M

y

y=0=m(0)

= 2m(0) + 0, (by (4.11) and(4.12)), (4.14)

i.e., m(0)=0 M

m(y) +

2M

2[m(y)]2 + 2 M

ym(y) +

2M

y 2

y=0=m(0)

=0,


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4.1. THE PERIOD-DOUBLING BIFURCATIONTHEOREM 39

which implies

2m(0) + 2M(m(0), 0)

y 2 = 2m(0) + 2[a1(0)a3(0)

+ 2a1(0)a22 (0) + a31 (0)a3(0)]= 2m(0) + 2 (1)[2(a3(0) + a22 (0))] =0,

m(0)= 2[a3(0) + a22 (0)]

= 2

1

3! 3g

y 3+

1

2!2g

y 2

2y=0=m(0)

= 2

1

3!3f

x3+

1

2!2f

x2

2

x=x0

=0

= 2

=0.

Therefore, neary=0, the function=m(y)has an expansion

=m(0) + m(0)y+m(0)2! y

2 +O(|y|3) (4.15)

=0

y2 +O(|y|3). (4.16)

Exercise 4.2 Verify that 3g2(0,0)

y 3 =3 2 M(0,0)

y 2 = 12=0.

We now check the stability of the period-2 points by computing (g 2)y

about y=0 and=0:

(g2)(, y)

y= (g

2)(0, 0)

y+

2(g2)(0, 0)

y 2 y+

2(g2)(0, 0)

y( 0)

+ 12

3(g2)(0, 0)

y 3 y2 + . (4.17)

But

(g2)(0, 0)

y=a 21 (0)=(1)2 =1, (4.18)

2

(g2

)(0, 0)y 2

=a1(0)a2(0) + a21 (0)a(0)= a2(0) + a2(0)=0, (4.19)

2(g2)(0, 0)

y=2a1(0)a1(0)= 2, (by(4.10)), (4.20)


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and1

2

3(g2)(0, 0)

y 3 = 1

2(12)= 6, (by Exercise4.2). (4.21)

Substituting(4.12)and (4.18)(4.21) into (4.17), we obtain

(g2)(m(y), y)

y=1 + (2)

y2

+ (6)y 2 +O(|y|3)=1 4y 2 +O(|y|3).

Therefore,

|1 4y 2| =1 4y 2 0, and so the period-2orbit isattracting;

|1 4y 2| =1 4y 2 >1, if


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4.2. SADDLE-NODE BIFURCATIONS 41

(a)

c

(b) 1

Figure 4.2: The graphs ofy=f (, x)=ex .(a) > e1;(b) =e1;(c) 0 < < e1.

In Fig.4.2(b), we see that the slope x

f(,x)is equal to 1 at the point of tangency wherex=1. There is a bifurcation of fixed points when=e1 becausef (,x)has changed behaviorfrom having no fixed points in Fig.4.2(a) to having two fixed points in Fig.4.2(c).This bifurcationis now analyzed in the following theorem.

Theorem 4.5 (Saddle-Node or Tangent Bifurcation) Assume thatf (, ) : R2

R isC2

satis-fying the following conditions:

f (0, x0)=x0and

xf (0, x0)=1,

2

x2f (0, x0)=0,

f (x0, 0)=0.


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(i) f (0, x0)=x0,(ii)

xf(,x)

=0x=x0

=1,

(iii) 2

x2f(,x)

=0x=x0

=0, and

(iv) f

(,x)

=0x=x0=0.

0

0

x0

x0

unstable

stable

(ii)

(i)

stable

unstable

Figure 4.3: Saddle-node bifurcation.

Then there exists a curve C : =m(x)of fixed points defined in a neighborhood ofx0such that0=m(x0), m(x0)=0, and

f (m(x), x)=x , m(x0)= 2f/x2

f/

x=x0=0

=0. (4.22)

The fixed points on Care either (i) stable for x > x0and unstable for x < x0, or (ii) stable for x < x0and unstable forx < x0. (See Fig.4.3.)

Proof. We follow Robinson [58, pp. 212213].To determine the fixed points, we define the function

G(,x )=f (,x) x=0.

ThenG(0, x0)=0. We have

xG(, x)

=0x=x0

=

f

x 1

=0x=x0

=0,


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If

2f (0,x0)

x 2 >0

, then

1 + 2f (0, x0)

x2 (x x0)

>1 x > x0,

if


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4.3. THEPITCHFORK BIFURCATION 45

x f(,x)=0x=0

=1,

g(,x)

=0x=0

=0,

2

x2f(,x)

=0x=0

=0,

3

x3f(,x)

=0x=0

=0.

Figure 4.4: Pitchfork bifurcation.

thenxg(, x) x=0=f (,x) x. Therefore, xis a fixed point of the mapf (,x). Since

xG(, x)

=0x=0

= x

g(, x)

=0x=0

= 12

2

x2f(,x)

=0x=0

=0,

we may not be able to solvex in terms oflocally near=0. However,

G(, x)

x=0=0

= g(,x)

=0x=0

=0,

so we have a curve C2 : =m(x)of fixed points near (, x)=(0, 0), such thatm(0)=0. Letus computem(0)andm(0). We haved

dxG(m(x), x)=0= g

m(x) + g

x.

Sinceg(0, 0)/=0,m(0)= g(0, 0)/x

g(0, 0)/=0. (4.24)

Differentiating again,

d2

dx 2G(m(x), x)=0=

2g

2[m(x)]2 + 2

2g

xm(x)

+ g

m(x) + 2

gx2

,

we obtain

m(0)= 2g(0, 0)/x

2

g(0, 0)/. (4.25)


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But 3

x3f(,x)= 3

x3[xg(,x)] =3 2g(,x)

x2 + x 3g(,x)

x3

and atx=0, =0, 2g(0, 0)

x2 = 1

3

3

x3f (0, 0)=0. (4.26)

From (4.25)and (4.26), we thus havem(0)=0. (4.27)

Combining (4.24) and (4.27), we see that the curveC2, locally, looks like a parabola near(0, 0)onthe(,x)-plane. C2opens to the left ifm(0) 0.

Finally, let us analyze stability of the bifurcated fixed points onC2near(, x)=(0, 0). Wehave

f (, x)

x

=m(x)

= f (0, 0)x

+ 2f (0, 0)

x2 (x 0) +

2f (0, 0)

x(m(x) 0)

+ 12!

3f (0, 0)

x3 (x 0)2 +

3f (0, 0)

x2 (x 0)(m(x) 0)

+ 12!

3f (0, 0)

2x(m(x) 0)2 +

=1 + 0 x+

g(0, 0)

m(0)x2 +O(x3)

+ 1

2

3

2g(0, 0)

x2 x2

+O(x3). (4.28)

But, by (4.25),g(0, 0)

m(0)=

2g(0, 0)

x2 , (4.29)

and by substituting (4.29) into(4.28), we obtain

f (, x)

x

=m(x)

=1 + 13

2g(0, 0)

x2 x2 +O(x3).

Therefore, fixed points onC2near(,x)=(0, 0)are attracting if2g(0, 0)/x

2 0.

Example 4.7 For the mapf (,x)= sin(x), we havef(,x)=xg(, x) whereg(,x)=

(x)

2

3! + (x)4

5! + (1)n(x)

2n

(2n + 1)!

.


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4.4. HOPF BIFURCATION 47

At(, x)

= 1 , 0, we havef (,x)

x=1, g(,x)

==0,

2f(,x)

x2 =0,

3f(,x)

x3 = 3 =0,

2g(, x)

x2 = 1

3 2


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where is a parameter; =

(),a=

a(),b=

b()are smooth functions of , satisfying 0< (0) < , a(0)=0, for=0.

It is easy to check that the origin, (x1, x2)=(0, 0), is a fixed point ofFFFfor all . At (0,0),the Jacobian matrix of the map FFF is

A(1 + )

cos sin sin cos

.

The two eigenvalues of matrix A are 1,2(1 + )ei. In particular,when =0,wehave |1,2| =1. Thus, the origin isnota hyperbolic fixed point; cf. Def.1.4. To facilitate the study of bifurcationof the system whenpasses=0, we rewrite FFFas a map of the complex plane:

forz=x1+ ix2, x1, x2 R,

FFF(z)=ei

z(1 + + d|z|2

)=z + cz|z|2

, (4.31)c=c()ei()d (), d ()a() + i b( ), =()(1 + )ei().

We look at the phase relation of (4.31): lettingz= ei with= |z|, we haveFFF(z)=ei(e i )[1 + + (a+ ib)2]

= [(1 + + a2)2 + b24]1/2ei(++) ,where

= sin1 b2

[(1 + + a2)2 + b24]1/2 .

Thus, in polar coordinates, system (4.31) becomes

GGG(,)= [1 + + a()2] + 4R ()

+ () + 2Q() , (4.32)where R() and Q () aresmoothfunctionsof(,).From (4.32),weknowthatthefirstcomponenton the RHS of (4.32) is independent of . Thus, we have achieved decouplingbetween and ,making the subsequent discussions on bifurcation more intuitive. With regard to the-variable, thetransformation (4.31) actually constitutes a 1-dimensional dynamical system:G(p) [1 + + a()2] + 4R().For this dynamical system, =0is a fixed point for any parameter value.When >0, the fixedpoint=0is unstable. When=0, the stability of=0is determined by the sign ofa(0):

(i) ifa(0) 0anda(0)


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With respect to the phase angle , the second component of the RHS of (4.32) shows that the

action of the map is similar to a rotation by an angle ()(but it depends on both and ).Summarizing the above, we see that for the 2-dimensional dynamical system (4.30),assuming

thata(0) 0. For this case, there is an unstable closed curve C

when >0, Hopf bifurcation happens whencrosses 0, and the curve C disappears.Next, we consider the following two-dimensional map:

FFF (x1, x2)=

cos sin sin cos

(1 + )

x1

x2+ (x21+ x22 )

a bb a

x1

x2+O(|x|4). (4.34)Similarly, to the conversion from (4.30) to(4.31), one can assert that(4.34)can be converted

to the formzF (z)=z + cz|z|2 +O(|z|4). (4.35)


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In comparison, system(4.34)contains some higher order terms than those of system (4.30). Even

though (4.34) is not locally topologically conjugate to (4.30), and the higher order terms in (4.34)do affect bifurcation phenomena, some special characters of Hopf bifurcation are preserved.

Lemma 4.9 The higher order term O(|x|4) in (4.34) does not affect the occurrence (ordisappearance)of the invariant closed close C and its stability. The local stability of the origin(0,0) and the bifurcation patterns remain the same.

Proof. The justification is lengthy. We refer to Kuznetsov[46, pp. 131136], for example.

Now, we consider a general planar map

fff: R

2

R

2, xxx

fff (xxx, ), xxx=

(x1, x2)R

2, R. (4.36)

We will prove that any planar map with the Hopf bifurcation property can be transformed into theform (4.34).

Assume thatfffis smooth, and at =0, fff (xxx , ) hasafixedpoint xxx= 000,i.e.,fff (000, 0)= 000.Theeigenvalues of the Jacobian matrix at (xxx,)=(000, 0) are 1,2=ei0 , for some 0 : 0 < 0 < . Bythe Implicit Function Theorem, for sufficiently small||, fffhas a one-parameter family of uniquefixed pointsxxx(), and the mapfffis invertible. By using translation, the fixed points xxx()can berelocated to the origin000. Thus, without loss of generality, we assume that for||small, xxx= 000is afixed point of the system. Thus, the map can be written as

xxxA()xxx+FFF (xxx,), (4.37)

whereA()is a 2 2matrix depending on , andFFF (xxx,)is a vector valued function with com-ponentsF1and F2such that their leading terms begin quadratically with respect to x1and x2inthe Taylor expansions ofF1and F2,and FFF (000, )= 000for sufficiently small ||. The Jacobian matrixA()has eigenvalues

1,2()=r()ei(), withr (0)=1, (0)=0. (4.38)

Set ()=r() 1. Then ()is smooth and (0)=0. Assume that (0)=0, then (locally)we can use in lieu offor the parametrization. Thus, we have

1()=()(1 + )e i(), 2()=(), (4.39)

where()is a smooth function of and (0)=0.

Lemma 4.10 Under the assumptions of (4.38) and (4.39) for||small, the map (4.37) can berewritten as

z()z + g(z, z, ), zC, R, (4.40)


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wheregis a smooth function with a (local)Taylors expansion

g(z,z,)=

k+2

1

k!!gk()zkz, k, =0, 1, 2, . . . .

Proof. Let qqq()be the eigenvector corresponding to eigenvalue ():

A()qqq()=()qqq(). (4.41)

ThenAT(), the transpose ofA(), has an eigenvector ppp()corresponding to eigenvalue():

AT()ppp()

=()ppp().

First, we prove thatppp,qqq ppp,qqqC2= 0,

wherep, qis defined as

p, q = p1q1+ p2q2 for p=(p1, p2)T, q=(q1, q2)T.

In fact, sinceAqqq= qqqand Ais a real matrix, we have Aqqq= qqq.Thus,

ppp,qqq =

ppp,1

Aqqq

=

1

ppp, A

qqq

=

1

ATppp,

qqq

= 1

ppp,qqq =

ppp,qqq.

Therefore, 1

ppp,qqq =0.

But= because for sufficiently small||, we have0 < () < . So we have

ppp,qqq =0.

Second, sinceAis a real matrix and the imaginary part ofis nonzero, the imaginary part ofqqqis also nonzero. By the above equality, we have

ppp, qqq =0.

By normalization, we can assume that

ppp, qqq =1.


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For any sufficiently small|

|, anyxxx

R2, there exists a unique z

C such that

xxx=zqqq() + z qqq(). (4.42)

(Due to the fact thatppp(),qqq()C2= 0, we can simply choosez= ppp(),xxxC2 .)From (4.37)and (4.41) through (4.42), for the complex variable z, we have

z()z + ppp(), F (zqqq() + z qqq(), )C2 . (4.43)

Denote the very last term in (4.43) as g(z, z,). Then we obtain

g(z,z,)=

k+21

k!!gk()zkz,

where

gk()= k+

zkzppp(), F (zqqq() + z qqq(),)|z=0,

fork+ 2, k, =0, 1, 2, . . .. Hence, (4.40)is obtained. The following three lemmas show that under proper conditions, we can convert the map from

the form (4.40) to the standard form (4.36).

Lemma 4.11 Assume thatei0 =1, e3i0 =1. Consider the map

z

z+

g20

2z2

+g

11zz

+g02

2z2

+O(

|z|

3), zC,

(4.44)

where =()=(1 + )e i(), gij= gij(). Then for|| sufficiently small, there exists a(locally)invertible transformation

z=w+ h202

w2 + h11ww+h02

2w2, (4.45)

such that (4.44) is transformed to

ww+O(|w|3), w C,

i.e., the quadratic terms O(

|z

|2)in (4.44) are eliminated.

Proof. It is easy to check that(4.45)is invertible near the origin, as

w=z

h20

2z2 + h11zz +

h02

2z2

+O(|z|3).


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With respect to the new complex variable w,(4.45) becomes

ww+ 12[g20+ ( 2)h20]w2 = [g11+ ( ||2)h11]ww

+ 12[g02+ ( )2h02]w2 +O(|w|3). (4.46)

As

2(0) (0)=ei0 (ei0 1)=0,|(0)|2 (0)=1 ei0 =0

(0)2 (0)=ei0 (ei30 1)=0,

for

|

|sufficiently small, we thus can let

h20= g20

2 , h11= g11

||2 , h02= g02

2 .

Hence, all the quadratic terms in (4.47) disappear. The proof is complete.

Remark 4.12

(i) Denote 0=(0). Then the conditions e i0 =1 and e 3i0 =1 in Lemma4.11mean that0=1, 30=1.The condition0=1 isautomaticallysatisfiedas 0=e i0 and 0 < 0 < .

(ii) From the transformation(4.45), we see that in the neighborhood of the origin, it is nearly anidentity transformation.

(iii) The transformation (4.45) generally alters the coefficients of the cubic terms.

Lemma 4.13 Assume thate2i0 =1, e4i0 =1. Consider the map

zz +g30

6z3 + g21

2z2z + g12

2zz2 + g03

6z3

+O(|z|4), (4.47)

where=()=(1 + )e i(), gij= gij(). For||small, the following transformation

z=w+

h30

6w3 + h21

2w2w+ h12

2ww2 + h03

6w3

(4.48)

converts (4.47) toww+ g21

2w2w+O|w|4), (4.49)

i.e., only one cubic term is retained in (4.49).


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Proof. The map is locally invertible near the origin:

w=z

h30

6z3 + h21

2z2z + h12

2zz2 + h03

6z3

+O(|z|4).

Substituting(4.48)into (4.47), we obtain

ww+ 16[g30+ ( 3)h30]w3 +

1

2[g21+ ( ||2)h21]w2w

+12[g12+ ( ||2)h12]ww2 +

1

6[g03+ ( 3)h03]w3

+O(|w|4).

If we seth30

= g30

3

, h12

= g12

||2

, h03

= g03

3

,

which is viable as the denominators are nonzero by assumption, as well as h21=0, then we obtain(4.49).

The terms g212

w2win (4.49) is called the resonanceterms. Note that its coefficient,g21/2, isthe same as the corresponding term in(4.47).

Lemma 4.14 (Normal form of Hopf bifurcation) Assume thateik0 =1fork=1, 2, 3, 4. Con-sider the map

zz +g20

2z2 + g11zz +

g02

2z2

+g30

6z3 + g21

2z2z + g12

2zz2 + g03

2z3

+O(

|z

|4), (4.50)

where =()=(1 + )e i(), gij= gij(), 0=()|=0.Thenthereexistsalocallyinvertibletransformation near the origin:

z=w+

h20

2w2 + h11ww+

h02

2w2

+

h30

6w3 + h12

2ww2 + h03

6w3

such that for||sufficiently small, (4.50) is transformed to

ww+ c1w2w+O(|w|4),

where

c1=

c1

()=

g20g11( 3 + 2)2(2 )( 1) +

|g11|21 +

|g02|22(2 ) +

g21

2. (4.51)

Proof. This follows as a corollary to Lemmas4.11and4.13.The valuec1in (4.51) can be obtainedby straightforward calculations.


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We can now summarize all of the preceding discussions in this section. As the map (4.34)

is(4.35)in essence, from Lemma4.9,we obtain the Hopf bifurcation theorem for general planarmaps as follows.

Theorem 4.15 (Hopf(NeimarkSacker) Bifurcation Theorem) For the 1-parameter family ofplanar maps

xxxf (xxx,),assume that

(i) When=0, the system has a fixed point xxx0= 000, and the Jacobian matrix has eigenvalues

1,2=ei0 , 0< 0 < .

(ii) r (0)=0, wherer()is defined through (4.38).(iii) eik0 =1, fork=1, 2, 3, 4.(iv) a(0)=1, wherea(0)=Re(ei0 c1(0)), withc1as given in(4.51).

Then whenpasses through 0, the system has a closed invariant curveCbifurcating from the fixedpoint xxx0= 000.

In applications, we often want to obtain the actual value ofa(0), which, from (4.51):

c1(0)

= g20(0)g11(0)(1 20)

2(20 0) +|g11(0)|2

1 0 + |g02(0)|2

2(2 0) +g21(0)

2

,

is

a(0)=Re

ei0 g21(0)2

Re

(1 2ei0 )e2i0

2(1 ei0 g20(0)g11(0)

12|g11(0)|2

14|g02(0)|2.

NOTES FOR CHAPTER 4The word bifurcation orAbzweigung(German) seems to have been first introduced by the celebratedGerman mathematician Carl Jacobi (1804-1851) [44] in 1834 in his study of the bifurcation of

the McLaurin spheroidal figures of equilibrium of self-gravitating rotating bodies (Abraham andShaw [1, p. 19], Iooss and Joseph [43, p. 11]). Poincare introduced the French word bifurcationin [57] in 1885. The bifurcation theorems studied in this chapter are of thelocal character, namely,local bifurcations, which analyze changes in the local stability properties of equilibrium points,periodic points or orbits or other invariant sets as system parameters cross through certain critical


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thresholds. The analysis of change of stability and bifurcation is almost always technical. No more

so than the case of maps when the governing system consists of ordinary differential equations oreven partial differential equations.

A partial list of reference sources for thestudy of bifurcations of maps,ordinary and partial dif-ferential equationsare Hale andKocak [31],IoossandJoseph[43], Guckenheimer andHolmes [30],Robinson [58], and Wiggins [68,69].


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57

C H A P T E R 5

Homoclinicity. LyapunoffExponents

5.1 HOMOCLINIC ORBITSThere is a very importantgeometricconcept, calledhomoclinic orbits, that leads to chaos.

Let pbe a fixed point of aC1

-mapf:f(p)=p.

Assume thatp is repelling so that|f(p)|> 1. Sincep is repelling, there is a neighborhood N(p)ofpsuch that

|f(x) f(p)| = |f(x) p|>|x p|, x N(p). (5.1)We denoteWuloc(p)the largest open neighborhood ofpsuch that (5.1) is satisfied. W

uloc(p)is called

thelocal unstable set ofp.

Definition 5.1 Let p be a repelling fixed point of a continuous mapf, and let Wuloc(p)be thelocal unstable set ofp. Letx0Wuloc(p). We say that x0ishomoclinicto pif there exists a positiveintegernsuch that

fn(x0)=p.We say thatx0isheteroclinicto pif there exists another different periodic point q such that

fm(x0)=q . See some illustrations in Fig.5.1(a) and (b).

Definition 5.2 A homoclinic orbit is said to benondegenerateiff(x)=0for all x on the orbit.Otherwise, it is said to bedegenerate. (See Fig.5.2.)

A nondegenerate homoclinic orbit will lead to chaos, as the following theorem shows.

Documents

Chen G., Huang Y. Chaotic Maps. Dynamics, Fractals, And Rapid Fluctuations (Morgan, 2011)(ISBN 159829914X)(O)(243s)_PNc