Dynamical Systems with Applications using MAPLE 978-0-8176-4150-4, 978-1-4899-2849-8

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STEPHEN LYNCH

Dynamical Systems with Applications using MAPLE

Springer Science+Business Media, LLC

Stephen Lynch Department of Computing and Mathematics Manchester Metropolitan University Manchester MI 5GD, UK s.lynch @doc.mmu.ac.uk

Library of Congress Cataloging·in·Publication Data Lynch, Stephen, 1964Dynamical systems with applications using MAPLE I Stephen Lynch. p. cm. Includes bibliographical references and index. ISBN 978-1-4899-2849-8 (eBook) ISBN 978-0-8176-4150-4 DOI 10.1007/978-1-4899-2849-8 1. Differentiable dynamical systems-Data processing. 2. Maple (Computer file) I. Title. QA614.8.L96 2000 515'.352-dc21

00-051905 CIP

AMS Subject Classifications: 34Axx, 34Cxx, 34Dxx, 37Exx, 37Gxx, 37Nxx, 58FIO, 58Fl4, 58F21, 78A25, 78A60, 78A97, 92Bxx, 92Exx, 93Bxx, 93Cxx, 93Dxx

Printedon acid-free paper. ©2001 Springer Science+Business Media New York Originally published by Birkhäuser Boston in 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publisher Springer Science+Business Media, LLC. except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the TradeMarks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 978-0-8176-4150-4

SPIN 10739889

Reformatted from author's files in U\'JEX2E by John Spiegelman, Philadelphia, PA. Cover design by Joseph Sherman, New Haven, CT.

9 8 7 6 5 4 3 2 I

Contents

Preface

xi

0 A Thtorial Introducdon to Maple 0.1 Tutorial One: The Basics (One Hour) . . . . . . . . . . . . 0.2 Tutorial Two: Plots and Differential Equations (One Hour) 0.3 Simple Maple Programs . Common Errors . 0.4 Maple Exercises . . . . . 0.5

1 2 4 6

1 Differential Equations 1.1 Simple Differential Equations and Applications 1.2 Applications to Chemical Kinetics 1.3 Applications to Electric Circuits . . 1.4 Existence and Uniqueness Theorem . 1.5 Maple Commands 1.6 Exercises . . . . . . . .

2 Linear Systems in the Plane 2.1 Canonical Forms . . . . . . . . . . . . . . . . . . . 2.2 Eigenvectors Defining Stable and Unstable Manifolds 2.3 Phase Portraits ofLinear Systems in the Plane 2.4 Maple Commands 2.5 Exercises . . . . . . . . . . . . . . . . . . . .

8 9

13 14 21 23 27 30 31

35 35 41 43

47 48

vi

3 Nonlinear Systems in the Plane Linearization and Hartman's Theorem 3.1 Constructing Phase Plane Diagrams 3.2 Maple Commands 3.3 Exercises . . . 3.4

Contents

51 51 53 61 62

4 Interacting Species Competing Species . . . . . . . . . . . . . . . . . 4.1 Predator-Prey Models . . . . . . . . . . . . . . . . 4.2 Other Characteristics Affecting Interacting Species 4.3 Maple Commands 4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . 4.5

65 65 68 74 74

5 Limit Cycles Historical Background . . . . . . . . . . . . . . . . . . 5.1 Existence and Uniqueness of Limit Cycles in the Plane 5.2 Nonexistence ofLimit Cycles in the Plane 5.3 Maple Commands 5.4 Exercises . . . . . . . . . . . . . . . . . . 5.5

77 77 80 85 88 88

6 Hamiltonian Systems, Lyapunov Functions, and Stability Hamiltonian Systems in the Plane . 6.1 Lyapunov Functions and Stability . 6.2 Maple Comlllands 6.3 Exercises . . . . . . . . . . . . . . 6.4

75

91 92

97 101 102

7 Bifurcation Theory Bifurcations ofNonlinear Systems in the Plane. 7.1 Mullistability and Bistability 7.2 Maple Commands . 7.3 Exercises . . . . . . . . . . . 7.4

105

8 Three-Dimensional Autonomous Systems and Chaos Linear Systems and Canonical Forms . 8.1 Nonlinear Systems and Stability . . . 8.2 The Rössler System and Chaos . . . . 8.3 The Lorenz Equations, Chua's Circuit, 8.4 and the Belousov-Zhabotinski Reaction Maple Commands 8.5 Exercises . . . . . . . . . . . . . . . . . 8.6

119 120 125 128

9 Poincare Maps and Nonautonomaus Systems in the Plane Poincare Maps . . . . . . . . . . . . . . . . . . . . . 9.1 Hamiltonian Systems with Two Degrees ofFreedom .. 9.2

106 112 115 115

132 139 140

143 144 150

Contents 9.3 9.4 9.5

vii Nonautonomous Systems in the Plane Maple Commands . . . . Exercises . . . . . . . . . . . . . . . .

152 163 165

10 Local and Global Bifurcations 10.1 Small-Amplitude Limit Cycle Bifurcations . 10.2 Melnikov Integrals and Bifurcating Limit Cycles from a Center . . . . . . 10.3 Homoclinic Bifurcations 10.4 Maple Commands . 10.5 Exercises . . . . . . . . .

169 170

11 The Second Part of David Hilbert's Sixteenth Problem 11.1 Statement of Problem and Main Results 11.2 Poincare Compactification 11.3 Map1e Commands . 11.4 Exercises . . . . . . . . . .

181 181 183 190 191

12 Limit Cycles of Lienard Systems 12.1 Global Results . 12.2 Local Results 12.3 Exercises . . . .

193 193 201 202

13 Linear Discrete Dynamical Systems 13.1 Recurrence Relations . . . . . 13.2 The Leslie Model . . . . . . . 13.3 Harvesting and Culling Policies . 13.4 Maple Commands . 13.5 Exercises . . . . . . . . . . . . . 14 Nonlinear Discrete Dynamical Systems 14.1 The Tent Map and Graphical Iterations 14.2 Fixed Points and Periodic Orbits . . . 14.3 The Logistic Map, Bifurcation Diagram, and Feigenbaum Number . 14.4 Gaussian and Henon Maps 14.5 Maple Commands . 14.6 Exercises . . . . . . 15 Complex Iterative Maps 15.1 Julia Sets and the Mandelbrot Set . 15.2 Boundaries of Periodic Orbits . 15.3 Maple Commands . 15.4 Exercises . . . . . . . . . . . .

174 176 177 179

205 . 206

210 214 218 219 223 224

231 236 244 249 251

255 256 259 263 264

Vlll

Contents

267

16 Electromagnetic Waves and Optical Resonators 16.1 Maxwell's Equations and Electromagnetic Waves 16.2 Historical Background of Optical Resonators . 16.3 The Nonlinear Simple Fiber Ring Resonator 16.4 Chaotic Attractors and Bistability . 16.5 Maple Commands . 16.6 Exercises . . . . . . . . . . . . . .

268 270 273 276 279 280

17 Analysis of Nonlinear Optical Resonators 17.1 Linear Stability Analysis . . 17.2 Instabilities and Bistability . 17.3 Maple Commands . 17.4 Exercises . . . . . . . . . . .

283 284 286 291 293

18 Fractals 18.1 Construction of Simple Examples . 18.2 Calculating Fractal Dimensions . 18.3 Maple Commands 18.4 Exercises . . . . . . . . . . . . .

295 295 301 307 310

19 Multifraetats 19.1 AMultifractalForm alism................... 19.2 Multifractals in the Real World and Some Simple Examples . 19.3 Maple Commands . 19.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 314 317 326 326

20 Controlling Chaos 20.1 Historical Background . . . . . . . . . 20.2 Controlling Chaos in the Logistic Map 20.3 Controlling Chaos in the Henon Map . 20.4 Maple Commands . 20.5 Exercises . . . . . . . . . . . . . . . .

329 330 334 336 341 343

21 Examination-Type Questions 21.1 Dynamical Systems with Applications . . . . . . . 21.2 Dynamical Systems with Applications Using Maple

347 347 350

22 Solutions to Exercises 22.0 Chapter 0 22.1 Chapter I 22.2 Chapter 2 22.3 Chapter 3 22.4 Chapter 4 22.5 Chapter 5

353 353 355 356 357 359 360

Contents 22.6 22.7 22.8 22.9 22.10 22.11 22.12 22.13 22.14 22.15 22.16 22.17 22.18 22.19 22.20

1x

Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20

............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................. ............................. ............................. ............................. ............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 361 363 364 365 365 367 367 369 370 371 371 371 372 373

References 375 Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Research Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Index

385

Preface

This book provides an introduction to the theory of dynamical systems with the aid of the Maple algebraic manipulation package. lt is written for both senior undergraduates and first-year graduale students. The firsthalf of the book deals with continuous systems using ordinary differential equations (Chapters 1-12) and the second half is devoted to the study of discrete dynamical systems (Chapters 13-20). (The author has gone for breadth of coverage rather than fine detail and theorems with proof are kept at a minimum.) The material is not clouded by functional analytic and group theoretical definitions, and so is intelligible to readers with a general mathematical background. Some of the topics covered are scarcely covered elsewhere. Most of the material in Chapters 9-12, 16, 17, 19, and 20 is at postgraduale Ievel and has been influenced by the author's own research interests. It has been found that these chapters are especially useful as reference material for senior undergraduate project work. The book has a very hands-on approach and takes the reader from the basic theory right through to recently published research material. An efficient tutorial guide to the Maple symbolic computation system has been included in Chapter 0. Students should be able to complete tutorials one and two in under two hours depending upon their past experience. The author suggests that the reader should save the relevant example programs listed throughout the book in separate files. These programs can then be edited accordingly when attempting the exercises at the end of each chapter. The Maple commands, programs and output can also be viewed in color over the Web at either

xii

Preface

http://www.birkhauser.com/cgi-win/ISBN/0-8176-4150-5 or Maple's applications site, http://www.maplesoft.com/appsl. Throughout the book, Maple is viewed as a tool for solving systems or producing eye-catching graphics. The author has used Maple V release 5.1 and Maple 6 in the preparation of the material. However, the Maple programs have been kept as simple as possible and should also run under later versions of the package. The first few chapters of the book cover some theory of ordinary differential equations and applications to models in the real world are given. The theory of differential equations applied to chemical kinetics and electric circuits is introduced in some detail. Chapter 1 ends with the existence and uniqueness theorem for the solutions of certain types of differential equation. The theory behind the construction of phase plane portraits for two-dimensional systems is dealt with in Chapters 2 and 3, and applications to modeling the populations of interacting species are discussed in Chapter 4. Limit cycles, or isolated periodic solutions, are introduced in Chapter 5. Since we live in a periodic world, these are the most common type of solution found when modeling nonlinear dynamical systems. They appear extensively when modeling both the technological and natural sciences. Hamiltonian (conservative) systems and stability are discussed in Chapter 6, and Chapter 7 is concerned with how planar systems vary depending upon a parameter. Bifurcation, multistability, and bistability are discussed. The reader is first introduced to the concept of chaos in Chapters 8 and 9, where three-dimensional systems and Poincare maps are investigated. These higher-dimensional systems can exhibit strange attractors and chaotic dynamics. Once again the theory can be applied to chemical kinetics and electric circuits; a simplified model for the weather is also briefty discussed. Both local and global bifurcations are investigated in Chapter I 0. The main results and statement of the famous second part of David Hilbert's sixteenth problern are listed in Chapter 11. In order to understand these results, Poincare compactification is introduced. The study of continuous systems ends with one of the authors specialities-Iimit cycles of Lienard systems. There is some detail on Lienard systems in particular in the first half of the book, but they do have a ubiquity for systems in the plane. Chapters 13-20 deal with discrete dynamical systems. Chapter 13 starts with a generat introduction to recurrence relations and iteration. Applications to population modeling and harvesting and culling policies is then investigated. Chaos in discrete systems is investigated and bifurcation diagrams are plotted in Chapter 14. The concept of universality is discussed for the first time. Complex iterative maps are introduced in Chapter 15. Julia sets and the now famous Mandelbrotset are plotted. As a simple introduction to optics, electromagnetic waves and Maxwell's equations are studied at the beginning of Chapter 16. Abriefhistory of nonlinear bistable optical resonators is discussed and the simple fiber ring resonator is

Preface

xiii

analyzed in particular. Chapters 16 and 17 are devoted to the study of these optical resonators and topics such as bifurcation, bistability, chaos, chaotic attractors, instabilities, linear stability analysis, multistability, and nonlinearity, which have already been dealt with in earlier chapters, are reviewed. Some simple fractals may be constructed using pencil and paper in Chapter 18, and the idea of fractal dimension is introduced. Fractals may be thought of as identical motifs repeated on ever reduced scales. Unfortunately, most of the fractals appearing in nature are not homogeneous but are more heterogeneous, hence the need for the multifractal theory given in Chapter 19. The final chapter is devoted to the new and exciting theory behind chaos control. For most systems, the maxim used by engineers in the past has been "stability good, chaos bad," but more and more nowadays this is being replaced with "stability good, chaos better." There are exciting and new applications to cardiology, Iaser technology, and space research, for example. This book is informed by the research interests of the author which are currently nonlinear ordinary differential equations, nonlinear optics and multifractals. Some references include recently published research articles. The prerequisites for studying dynamical systems using this book are undergraduale courses in linear algebra, real and complex analysis, calculus and ordinary differential equations; a knowledge of a computer language such as Fortran or Pascal would be beneficial but not essential. I would like to express my sincere thanks to David Chillingworth (Southampton), Colin Christopher (Plymouth), Yibin Fu (Keele), Lida Nejad (MMU), Tito Toro O.B.E. (MMU), Alan Steele (Nortel, Canada), Edward Vrscay (Waterloo, Canada), and the referees for their constructive comments on the first draft of the book. My thanks also go to Caroline Graf (Birkhäuser), Tom Grasso (Birkhäuser), and Paul Goossens (Maple). Special thanks go to Elizabeth Loew for allher help with the cover, production, and manufacturing of my book, as weil as to Ann Kostant (Executive Editor, Mathematics and Physics, Birkhäuser). I am especially grateful to John Spiegelman for his care and attention to the many small but important details that bad been overlooked, as weil as the beautiful typesetting of my book. It was a pleasure to work with him. Finally, thanks to my family and especially Gaynor for all their Iove and support.

Stephen Lynch

0 A Tutorial Introduction to Maple

Aims and Objectives • To provide a tutorial guide to the Maple package. • To give practical experience in using the package. • To promote self-help using the on-Iine help facilities. On completion of this chapter, the reader should be able to • use Maple as a mathematical tool; • produce simple Maple programs; • access some Maple commands and programs over the Web. lt is assumed that the reader is familiar with either the Windows or Unix environment. Commands listed in Sections 0.1 and 0.2 have been chosen to allow the reader to become familiar with Maple in a few hours. These tutorial sheets have been used with great success over a number of years with both mathematics and engineering undergraduate students. Experience has shown that the Maple worksheets can be completed in under two hours, after which students are able to adapt the commands to tackle their own problems. This method of teaching works weil with computer Iabaratory class sizes of no more than 20 students to one staff member. Section 0.3 gives a brief introduction to programming with Maple.

0. A Thtorial Introduction to Maple

2

If any problems result, there are several options. For example, there is an excellent help browser in Maple, the 10 most common errors are listed in Section 0.4, and Maple commands and programs with the respective output from this text can be found on the Web at http://www.birkhauser.com/cgi-win/ISBN/0-8176-4150-5 or http://www.maplesoft.com/apps/. The Maple worksheets on the Web may be edited and copied. Remember to save your Maple flies at regular intervals. You could Iabel your first file as tutl.mws, for example.

0.1

Tutodal One: The Basics (One Hour)

There is no need to copy the comments; they are there to help you.

Click on the Maple icon and copy the command after the > prompt. Maple Commands > #

>

This is a comment

1+2-3;

Comments # Helps when writing # programs. #

Simple addition and

# subtraction. >

2*3/7;

>

2*6+3A2-4/2;

>

(5+3)*(4-2);

>

sqrt(lOO);

> n1:=10: >

lprint('nl:=',nl):

i Multiplication and # division.

# The square root. # The colon suppresses # the output. # Use the • character # for quotes.

> nlA(-1);

# Negative powers.

> sin(Pi/3);

# Use capital P for Pi.

>

y:=sin(x)+3*xA2;

# Equations and # assignments.

>

evalf(sin(Pi/3));

# Evaluate as a floating # point number.

3

0.1. Totorial One: The Basics (One Hour) > diff (y, X) ;

# Differentiate y with # respect to x.

> y:='y':

# Set y back equal to y. # Partial differentiation.

> int(cos(x),x);

# Integration with # respect to x.

> int(x/(xA3-l),x=O .. l);

# Definite integrals.

> int(l/x,x=l .. infinity);

# Improper integrals.

> convert(1/((s+1)*(s+2) ),parfrac,s);

# Split into partial # fractions.

> expand(sin(x+y));

# Expansion.

> factor(xA2-yA2);

# Factorization.

> limit((cos(x)-1)/x,x=O) ;

# The limit as x goes # to zero.

> z1:=3+2*I;z2:=2-I;

# Complex numbers. Use # I NOT i.

> z3:=z1+z2; > z4:=z1*z2/z3; > modzl:=abs(z1);

# Modulus of a complex # number.

> evalc(exp(I*z1));

# Eva1uate as a complex # number.

> solve({x+2*y=1,x-y=3},{ x,y});

# Solve two simultaneous # equations.

> fsolve(x*cos(x)=O,x=7 .. 9);

# Find a root in a given # interval.

> S:=sum(iA2,i=1 .. n);

# A finite sum.

> ?linalg

# Open a help page.

> with (linalg):

# Load the linear # algebra package.

> A:=matrix( [ [1,2], [3,4]]); > B:=matrix([[1,0], [-1.3]]);

# Defining 2 by 2 # matrices.

> evalm(BA(-1));

# Matrix inverse.

> C:=evalm(A+2*B);

# Evaluate the new

4

0. A Thtorial Introduction to Maple # matrix.

> AB:=evalrn(A &* B);

# Matrix rnultiplication.

> Al: =rnatrix ( [ [ 1, 0, 4] , [ 0, 2, 0] , [ 3, 1, -3]]) ; > det (All;

# The deterrninant.

> eigenvals(Al);

# Gives the eigenvalues # of Al.

> ?eigenvects

# Shows how the eigen# vectors are displayed.

> eigenvects(Al);

# Gives the eigenvectors # of Al.

> # Use of the help browser - one option. > ?interp

# Open a help page for # interpolation.

>??interp

# List the syntax for # this cornmand.

>???interp

# List sorne exarnples.

> # End of Tutorial One.

Exit the Maple worksheet by clicking on the File and Exit buttons, but remernher to save your work.

0.2 Tutorlai Two: Plotsand Differential Equations (One Hour) There is no need to copy the comments, they are there to help you.

Click on the Maple icon and copy the command after the > prompt. Maple Commands

Comments

> ?plot

# Open a help page.

> with(plots):

#

> plot(cos(2*x),x=0 .. 4*Pi);

# Plot a trigonornetric # function.

Load the plots package.

> plot(x*(xA2-l),x=-3 .. 3,y=-10 .. 10, > title='A cubic polynornial'); # Plot a cubic polynornial # and add a title. > plot(tan(x),x=-2*Pi .. 2*Pi,y=-10 .. 10,

0.2. Thtorial Two: Plotsand Differential Equations (One Hour) > discont=true);

5

# Plot a function with

# discontinuities. > plot({x*cos(x),x-2},x=-5 .. 5);

# Plot two curves on one # graph.

> cl:=plot(sin(x),x=-2*Pi .. 2*Pi, > linestyle=l):

> c2:=plot(2*sin(2*x-Pi/2),x=-2*Pi .. 2*Pi, > linestyle=3): >

display({cl,c2});

> > > >

points:=[[n,sin(n)]$n=l .. l0]: pointplot(points,style=point, symbol=circle); pointplot(points,style=line);

> >

implicitplot(yft2+y=xft3-x,x=-2 .. 3, # Implicit plots. y=-3 .. 3) i

> >

animate(sin(x*t),x=-4*Pi .. 4*Pi,t=O .• l, color=red); i 2-D animation.

# Plot points and lines # joining the points on # two separate graphs.

> plot3d(sin(x)*exp(-y),x=O .. Pi,y=0 .. 3, > axes=boxed) ; i 3-D plots. You can i rotate the figure # with the left mouse

# button. > cylinderplot(z+3*cos(2*theta),

> theta=O .. Pi,z=0 .. 3); > animate3d(t*yft2/2-xft2/2+xft4/4,x=-2 .. 2, > y=-2 .. 2,t=0 .. 2); #3-D animation.

> ?DEtools

# Open a help page.

> with(DEtools):

# Load the differential # equations package.

>

dsolve(diff(y(x),x)=x,y(x));

# Solve a differential i equation.

> dsolve({diff(v(t),t)+2*t=O,v(l)=5},

> v(t)); >

Solve an initial value # problem.

i

dsolve(diff(x(t),t$2)+8*diff(x(t),t)

> +25*x(t)=O,x(t));

i

Solve second-order

i differential equations. > >

dsolve(diff(x(t),t$2)+8*diff(x(t),t) +25*x(t)=t*exp(t),x(t));

6

0. A Thtorial Introduction to Maple

>

deqn:=diff(y(x),x$2)=xA3*y(x)+1;

> >

DEplot(deqn,y(x),x=-3 .. 2, [[y(0)=0.5,D(y) (0)=1]]);

# Plot a solution curve.

> # > #

Differential equations will be considered in more detail in Chapter 1.

> #

End of Tuterial Two.

0.3

Simple Maple Programs

Programming in Maple is much simpler than programming in some other languages. The Maple Ianguage contains powerful commands, which means that some complex programs may contain only a few lines of code. Of course, the only way to learn programming is to sit down and try it yourself. The aim of this section is to introduce simple programming techniques by example. The programs are kept short to aid in understanding; the output is also included.

Procedures. You can create your own procedures. For example, the command norm3d below gives the norm of a three-dimensional vector. > # The norm of a vector in three-dimensional space. > norm3d:=proc(a,b,c) > sqrt(aA2+bA2+cA2); end; > >

norm3d := proc(a, b, c) sqrt(aA2 + bA2 + cA2) end > norm3d(3,4,5);

5 sqrt(2)

The for••do.. od loop. This type of command is used in most languages. > # A program to sum the natural numbers from 1 to imax. > # Note that the do must be ended with an od: > i:='i':total:=O: > for i from 0 to 100 do > total:=i+total: > od: > total; 5050

Conditional statements. If, then, elif, eise, etc. > # A simple program- note that if must be ended with a fi:

> p:=4: > if p >

elif p>=2 then lprint('p is not less than 2'): fi:

0.3. Simple Maple Programs

7

p is not less than 2

Arrays and sequences. Set up an array; F in this case can hold up to 10001 elements. This isasimple program to evaluate the first fifteen terms of the Fibonacci sequence. > #

The Fibonacci sequence.

> F:=array(O .. 10000) :F[OJ :=O:F[1]:=1:imax:=14:

for i from 2 to imax do

>

> F[i] :=F[i-1]+F[i-2]: > od: > > seq(F[i],i=O .. imax);

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377

To conclude this section, some options within plots will be highlighted.

Display. This command can be used to show multiple plots with text. See the examples below. >

# A program to p1ot two functions on one graph.

> with(plots): > p1:=plot(xA2,x=-4 .. 4,co1or=blue): > p2:=p1ot(4-xA2,x=-4 .. 4,color=red): >

t1:=textplot([2.6,6, •y=xA2'],a1ign=RIGHT):

> t2:=textp1ot([-3,-6, 'y=4-xA2'],align=RIGHT): >

disp1ay({p1,p2,t1,t2},font=[TIMES,ROMAN,20],tickmar ks=[3,3]);

\ I

ｾﾷ＠

··-

t-2

-4

i

10

,./ I

ｾ］ＴＭｸＲ＠

/

-10

Figure 0.1: Multipleplot with text. (Unfortunately, the colors are missing here.) > #

Two so1ution curves on one graph.

> with(DEtoo1s):with(plots): > deqn1:=diff(x(t),t$2)=-2*diff(x(t),t)-25*x(t);

0. A Thtorial Introduction to Maple

8 > > > > > > > >

pl:=DEplot(deqnl,x(t),t=O .. lO, [[x(O)=l,D(x) (O)=O]],stepsize=O.l, linestyle=l,linecolor=black): deqn2:=diff(x(t),t$2)=-25*x(t); p2:=DEplot(deqn2,x(t),t=O .. lO, [[x(O)=l,D(x) (O)=O]],stepsize=O.l, linestyle=7,linecolor=black): tl:=textplot([lO,l, 'Harmonie motion'],align=RIGHT): t2:=textplot([10,0, 'Damped motion'],align=RIGHT): display({pl,p2,tl,t2},font=[TIMES,ROMAN,15],labels=['t', 'x'));

1

\ X

0.5

fl

ｾI ＠ II'

u

I'-'

,i

-o .5

-1

1'-'

ｾ＠ ｾ＠ ｾ＠

I

Hannanic-

r

I

I

I

I ｾ＠

1

I

I

0

I

\

Figure 0.2: Solution curves for differential equations deqnl and deqn2.

0.4

Common Errors

Do not forget to end the command with either a semicolon (to see the output) or a colon (to suppress the output). Remember to check the help pages within Maple and the Web sites given at the beginning of the chapter if this page does not help you. The Error

The Command and Error Message

1. Omission of a bracket.

>(5+3)*4-2); syntax error: •) • unexpected

2. Omission of a colon.

>nl=lO: No error message.

3. Negative powers.

>nlA-1; syntax error: •-• unexpected

0.5. Maple Exercises

9

4. Use a capital P for the nurober Pi and a small p for the letter.

>sin(pi/3); No error message.

5. Quotes - ' or '

>lprint ( 'nl: =', nl); syntax error: ':=' unexpected

6.

Omission of multiplication sign.

>y:=sin(x)+3x"2; syntax error missing operator or ';'

7.

Omission of a dot.

>int(x/(x"3-l),x=O.l); error (in int) wrong nurober (or type) of arguments

8. Complex nurobers - use I not i.

>zl:=3+2*i; No error message.

9. Brackets.

>A:=matrix( [1, 2), [3, 4)); error (in matrix) 1st and 2nd arguments must be nonnegative integers

10. Matrix multiplication.

>AB:=evalm(A*B); error (in evalm/evaluate) use the &* operator for matrix/vector multiplication

The programs throughout the book should all compile under both Maple V and Maple 6. If you experience difficulties with Maple 6 see the updstrc text files, which are located in the directory in which Maple 6 is installed.

0.5

Maple Exercises

1. Evaluate the following: (a) 12 + 4- 5;

(b) 210;

(c) sin(O.l); (d) ((2 + 3)(4- 3(9- 5))).

2. Find the derivatives of the following functions:

= 3x 3 + 2x 2 - 5; y = J 1 + x4; y = ex sinx cosx.

(a) y (b)

(c)

3. Evaluate the following definite integrals:

10

0. A Totorial Introduction to Maple (a)

J}=0 3x 3 + 2x 2 -

(b)

fxC:I ｾ＠

5 dx;

dx;

(c) ｦｾｯ･Ｍｸｬ＠

dx.

4. Evaluate the following Iimits:

(c) lim x-+rr cosx+l x-rr .

5. Given that ZI (a)

= I + i, zz = 2 + 3i, and Z3 = 4- 2i, evaluate the following:

z1+zz-z3;

(b) illl· '

ZJ

(c) ez';

(d) ln(ZJ);

(e) sin(z3). 6. Load the linear algebra package by typing with(linalg):. Given that

A

=(

-1)

1 2 1 0 3 -1

0 2

'

B=

( 1 2 3) 1 0

1 2 1 2

,

c=(

2 0 4

1

1

1 ) -1 '

2

2

determine the following: (a) 2A- BC; (b)

s- 1;

(c) the eigenvalues and eigenvectors of C. 7. Load the plots package by typing with(plots):. Graph the following: (a) y

= 3x 3 + 2x 2 -

(b) y

= e-x

2

for -5 :5 x :5 5;

(c) x 2 - 2xy - y 2

(d) z

5;

= 1;

= 4x 2eY- 2x 4 -

e 4Y

for -3 :5 x :53 and -1 :5 y :51.

8. Load the differential equations package by typing with(DEtools):. Solve the following differential equations: (a)

*= fy.

given that y(l)

= 1;

0.5. Maple Exercises (b) ｾ＠

+ Ｕｾ＠

11

+ 6x = 0, given that x(O) = 1 and i(O) = 0.

9. Carry out 100 iterations on the recurrence re1ation Xn+i

given that (a) xo case.

= 4x"(l -

x"),

= 0.2 and (b) xo = 0.2001. List the finallO iterates in each

10. Type ?while to read the help page on the while command. Use a while-do-od loop to program Euclid's algorithm for finding the greatest common divisor of two integers. Hint: Ese the irem command. Use the program to find the greatest common divisor of 12348 and 14238.

Recommended Textbooks [1] K. M. Heal, M. Hansen, and K. Rickard,Maple 6 Learning Guide, Waterloo Maple, Toronto, 2000. [2] M. B. Monagan, K. 0. Geddes, K. M. Heal, G Labahn, S. M. Vorkoetter, and J. McCarron, Maple 6 Programming Guide, Waterloo Maple, Toronto, 2000. [3] M. Abell and J. Brasellon, Maple V by Example, Second ed., Academic Press, New York, 1998. [4] K. M. Heal, M. Hansen, and K. Rickard, Maple V Learning Guidefor Release 5, Springer-Verlag, Berlin, New York, Heidelberg, 1997. [5] M. Kofter, Maple: An lntroduction and Reference, Addison-Wesley, Reading, MA, 1997. [6] M. B. Monagan, K. 0. Geddes, G Labahn, and S. Vorkoetter, Maple V: Programming Guide, Springer-Verlag, Berlin, New York, Heidelberg, 1996.

1 Differential Equations

Aims and Objectives • To review basic methods for solving some differential equations. • To apply the theory to simple mathematical models. • To introduce an existence and uniqueness theorem. On completion of this chapter, the reader should be able to • solve certain first- and second-order differential equations; • apply the theory to chemical kinetics and electric circuits; • interpret the solutions in physical terms; • understand the existence and uniqueness theorem and its implications. Basic theory of ordinary differential equations (ODEs) and analytical methods for solving some types of ODEs are reviewed. This chapter is not intended to be a comprehensive study of differential equations, but more an introduction to the theory that will be used in later chapters. Most of the material will be covered in first- and second-year undergraduate mathematics courses. The differential equations are applied to all kinds of models, but this chapter concentrates on chemical kinetics and electric circuits in particular. The chapter ends with the existence and uniqueness theorem and some analysis.

1. Differential Equations

14

1.1

Simple Differential Equations and Applications

Definition 1. A differential equation that involves only one independent variable is called an ordinary differential equation (or ODE). Those involving two or more independentvariables are called partial differential equations. This chapter will be concemed with ODEs only.

The subject of ordinary differential equations encompasses analytical, computational, and applicable fields of interest. There are many textbooks written from the elementary to the most advanced, with some focusing on applications and others concentrating on existence theorems and rigorous methods of solution. This chapter is intended to introduce the reader to both branches of the subject.

Separable Differential Equations. Consider the differential equation dx dt = f(t, x),

(1.1)

and suppose that the function f(t, x) can be factored into a product f(t, x) = g(t)h(x), where g(t) is a function oft and h(x) is a function of x. If f can be factored in this way, then equation ( 1.1) can be solved by the method of separation of variables. To solve the equation, divide both sides by h(x) to obtain I dx h(x) dt = g(t); integration with respect to t gives

f ｨＨｾＩ＠

ｾ［＠

f

dt =

g(t)dt.

Changing the variables in the integral gives

f ｨｾ［Ｉ＠

=

f

g(t)dt.

An analytic solution to ( 1.1) can be found only if both integrals can be evaluated. The method can be illustrated with some simple examples.

Example 1. Solve the differential equation i

= -f.

Solution. The differential equation is separable. Separate the variables and integrale both sides with respect to t. Therefore,

and so

f ｘｾ［＠ J

f J

dt = -

xdx = -

t dt,

t dt.

1.1. Simple Differential Equations and Applications

15

Integration of both sides yields

where r 2 is a constant. There are an infinite nurober of solutions. The solution curves are concentric circles of radius r centered at the origin. There are an infinite nurober of solution curves that would fill the plane if they were alt plotted. Three such solution curves are plotted in Figure 1.1.

Figure 1.1: Three solution curves for Example 1.

Example 2. Solve the differential equation x =

-!,;. :;c

Solution. The differential equation is separable. Separate the variables and integrate both sides w.r.t. t to give

Integration of both sides yields

where C is a constant. Six of an infinite nurober of solution curves are plotted in Figure 1.2.

Example 3. The population of a certain species of fish living in a large Iake at timet can be modeled using Verhulst's equation, otherwise known as the logistic equation, dP

dt =

P(ß- BP),

16

1. Differential Equations 4

-4

Figure 1.2: Six solution curves for Example 2. where P(t) is the population of fish measured in tens of thousands, and {J and ｾ＠ are constants representing the birth and death rates of the fish living in the Iake, respectively. Suppose that fJ = 0.1' ｾ＠ = 1o-3 and the initial population is 50 X 10". Solve this initial value problern and interpret the results in physical terms. Solution. Using the methods of separation of variables gives

f ｐＨ￟､ｾ＠

ｾｐＩ＠

=

f

dt.

The solution to the integral on the left may be determined using partial fractions. The generat solution is

ｴｮｪｻｊ｟ｐｾｰｬ＠

=ßt+C,

or {J - ｾ＠ + k{Je-ßt •

P(t)-

computed using Maple, where C and k are constants. Substituting the initial conditions, the solution is 100 P(t) = 1 + e-O.It. Thus as time increases, the population of fish tends to a value of 100 x 10". The solution curve is plotted in Figure 1.3. Note the following: • The quantity ｾ＠ is the ratio of births to deaths and is called the carrying capacity of the environment. • Take care when interpretlog the solutions. This and similar continuous models only work for large species populations. The solutions give approximate

1.1. Simple Differential Equations and Applications

17

)()()

90 80 p

70

60

Figure 1.3: Solution curve for the initial value problern in Example 3. numbers. Even though time is continuous, the population size is not. For example, you cannot have a fractionalliving fish, so population sizes have to be rounded out to whole numbers in applications. • Discrete models can also be applied to population dynamics (see Chapter 13).

Exact Differential Equations. A differential equation of the form M(t, x)

(1.2)

- =0 + N(t, x )dx dt

is said tobe exact if there exists a function, say F(t, x), with continuous second partial derivatives such that

aF

-

ar

= M(t x)

·

and ｾ＠

aF

= N(t,x).

Such a function exists as long as

aM

aN

-=-,

ax

ar

and then the solution to (1.2) satisfies the equation F(t,x) = C,

where C is a constant. Differentiate this equation with respect to t to obtain ( 1.2).

1. DifTerential Equations

18 Example 4. Solve the differential equation

9- 121- Sx dx dl = 51 + 2x - 4 · Solution. Inthiscase,M(t,x) = -9+121+5xandN(I,x) =51+2x-4.Now

aM

aN

-=-=5,

ax

at

and integration gives the solution F(t, x) = x 2 +61 2 +Stx -91 -4x = C. There are an infinite number of solution curves, some of which are shown in Figure 1.4.

Figure 1.4: Some solution curves for Example 4. Homogeneous Differential Equations. Consider differential equations of the form (1.3) Substitute v =

dx = dt

f (::). I

f into (1.3) to obtain d dt (vt) = f(v).

Therefore,

dv v + t dt = f(v),

1.1. Simple Differential Equations and Applications andso

dv

19

f(v)- v

dt =

t

which is separable. A complete solution can be found as long as the equations are integrable, and then v may be replaced with

f.

Example 5. Solve the differential equation

dx dt

=

t -x t +x

Solution. The equation may be rewritten as

dx

Let v =

1-:!

-=--'. dt 1+ f

(1.4)

f. Then (1.4) becomes dv 1- 2v- v2 dt = t(l + v) ·

This is a separable differential equation. The general solution is given by

x 2 +2tx

- t2 =

C,

where C is a constant. Some solution curves are plotted in Figure 1.5.

Figure 1.5: Some solution curves for system (1.4).

20

1. Differential Equations

Linear Differential Equations. Consider differential equations of the form

dx dt

(1.5)

+ P(t)x =

Q(t).

Multiplying by an integrating factor, say J(t), equation (1.5) becomes

dx Jdt+JPx=JQ.

(1.6)

Find J such that ( 1.6) can be written as

d dx dJ dt(Jx)=J dt +x dt =JQ. In order to achieve this, set

dJ -=JP; dt

integrale to get J(t)=exp(/ P(t)dt). Thus the solution to system ( 1.5) may be found by solving the differential equation d -(Jx) dt

= JQ

as long as the right-hand side is integrable.

Example 6. A chemical company pumps v Iiters of solution containing mass m grams of solute into a large Iake of volume V per day. The inftow and outftow of the water is constant. The concentration of solute in the Iake, say a, satisfies the differential equation (1.7)

da v m -+-a=-. dt V V

Determine the concentration of solute in the Iake at time t assuming that a = 0 when t = 0. What happens to the concentration in the long term?

Solution. This is a linear differential equation, and the integrating factor is given by J =exp(/

ｾ､ｴＩ＠

］･ｾＮ＠

Multiply (1.7) by the integrating factor to obtain

d(v')

vrm

dt eVa = eV V"

21

1.2. Applications to Chemical K.inetics Integration gives

m

VI

cr(t) = - - ke-v, V

where k is a constant. Substituting the initial conditions, the final solution is cr(t)

As t ｾ＠

=: (1- e-1?).

oo, the concentration settles to ｾ＠ gl- 1•

1.2 Applications to Chemical Kinetics Even the simplest chemical reactions can be highly complex and difficult to model. Physical parameters such as temperature, pressure, and mixing, for example, are ignored in this text, and differential equations are constructed that are dependent only on the concentrations of the chemieals involved in the reaction. This is potentially a very difficult subject and some assumptions have to be made to make progress. The Chemical Law of Mass Action. The rates at which the concentrations of the various chemical species change with timeareproportional to their concentrations. Consider the following example, in which one molecule of hydrogen reacts with one molecule of oxygen to produce two molecules of hydroxyl (OH):

Suppose that the concentration of hydrogen is [H2] and the concentration of oxygen is [02]. Then from the chemical law of mass action, the rate equation is given by Rate= k[H2H02], where k is called the rate constant, and the reaction rate equation is d[OH]

--;[! = 2k[H2][02]. Unfortunately, it is not possible to write down the reaction rate equations based on the stoichiometric (balanced) chemical equations alone. There may be many mechanisms involved in producing OH from hydrogen and oxygen in the above example. Even simple chemical equations can involve a large number of steps and different rate constants. Suppose in this text that the chemical equations give the rate-determining steps. Suppose that species A, B, C, and D have concentrations a(t), b(t), c(t), and d(t) at timet and initial concentrations ao, bo, co, and do, respectively. Table 1.1 lists some reversible chemical reactions and one of the corresponding reaction rate equations, where k 1 and kr are the forward andreverserate constants, respectively.

1. Differential Equations

22 Chemical reaction

The reaction rate equation for one species may be expressed as follows:

= ktab = kt(ao- c)(bo- c) b = kt(ao- 2b) 2 - k,b b = kt(ao- ｾＩＭ k,b2 i: = kt(ao- c)- k,(bo + c)(co + c) i: = kt(ao- c)(bo- c)- k,c i: = kt 0, and these critical points are unstable. When n is odd,

f' (xn)

= -1 < 0, and these critical points are stable. (b) There is one critical point at xo = 0 and f' (x) = 2x in this case. Now f' (0) = 0 and f" (0) = 2 > 0. Therefore, xo is attracting when x < 0 and repelling when xo > 0. The critical point is called semistable. (c)There is one critical point atxo = 0. Now f'(O) = -1 < 0, and therefore the critical point at the origin is stable. The theory of linear autonomous systems of ODEs in two dimensions will be discussed in the next chapter.

1.5

Maple Commands

The Maple commands below may be edited to produce solutions and diagrams for all of the examples and exercises in Chapter 1. > with(DEtools):with(plots):

> # Assign a differential equation. > deqn1:=diff(y(x),x)=-x/y(x);

d

X

y(x)

deqnl :=

y(xl

dx

> # Solve the differential equation > dsolve(deqnl,y(x));

2

2

y(x) >

sqrt(-x

+_Cl), y(x)

-sqrt(-x

+_Cl)

ｃｬＺ］ｸｾＲＫｹ＠

> implicitplot({Cl=l,Cl=4,Cl=9},x=-4 .. 4,y=-4 .. 4,numpoints=1000,

> color=blue,scaling=CONSTRAINED); > # See Figure 1.1. > # Population differential equation

> deqn2:=diff(P(t),t)=P(t)*(100-P(t))/1000; d

P(t) = 1/1000 P(t)

deqn2 := dt

> dsolve({deqn2,P(0)=50},P(t));

(100 - P(t))

31

1.6. Exercises

P(t)

1 100 ----------------1 + exp(- 1/10 t)

> dsolve({deqn2,P(0)=150),P(t));

P(t)

> > > >

1 100 --------------------1 - 1/3 exp(- 1/10 t)

p1:=plot(100/(1+exp(-0.1*t)),t=0 .. 70,color=blue): p2:=plot(100/(1-(1/3)*exp(-0.1*t)),t=0 .. 70,color=red): display( {p1,p2}); # See Figure 1.3.

> # Chemical kinetics differential equation > a:=4:b:=1:c:='c':k:=0.00713: > deqn:=diff(c(t),t)=k*(a-c(t))A2*(b-c(t)/2);

d

2

c

deqn :=

(t)

.00713 (4- c(tll

(1- 1/2 c(t))

dt > DEplot(deqn,c(t),t=0 .. 400, [[c(O)=O)),stepsize=0.01,c=0 .. 2.5, > linecolor=black,font=[TIMES,ROMAN,25)); > # See Figure 1.6.

1.6 Exercises 1. Sketch some solution curves for the following differential equations: (a)

(b)

*=-l· Ｊ］ｾﾷ＠ *= =.i.fx; X'

X

X '

ｾ＠ (d) dx (e) ｾＭ

X'

(c)

(f)

xy . xr:i7"'

* ?· di-

=

2. Fossils are often dated using the differential equation

dA = -ctA, dt

-

where A is the amount of radioactive substance remaining, a is a constant, and t is measured in years. Assuming that a = 1.5 x w- 7 , determine the age of a fossil containing radioactive substance A if only 30% of the substance remains.

32

1. Differential Equations 3. Write down the chemical reaction rate equations for the reversible reaction equations (a) A

+ B + C ;:::= D,

(b) A

+ A + A ;:::= A3,

given that the forward rate constant is k f and the reverse rate constant is kr in each case. Assurne that the chemical equations are the rate-determining steps.

4.

(a) Consider a series resistor-inductor circuit with L = 2H, R = 100, and an applied e.m.f. of E = IOOsin(t). Use an integrating factor to solve the differential equation, and find the current in the circuit after 0.2 seconds given that I (0) = 0. (b) The differential equation used to model a series resistor-capacitor circuit is given by Q dQ Rdt+ C = E, where Q is the charge across the capacitor. If a variable resistance 1/(5 + t)O and a capacitance C 0.5F are connected in series R with an applied e.m.f., E = IOOV, find the charge on the capacitor given that Q(O) = 0.

=

=

5. A forensie scientist is called to the scene of a murder. The temperature of the corpse is found tobe 75° F; one hour later the temperature has dropped to 70° F. Ifthe temperature ofthe room in which the body was discovered is a constant 68° F, how long before the first temperature reading was taken did the murder occur? Assurne that the body obeys Newton's Law of Cooling:

dT

-dt = ß(T- TR), where T is the temperature of the corpse, temperature.

ß is a constant, and TR is room

6. The differential equation used to model the concentration of glucose in the blood, say g(t), when it is being fed intravenously into the body is given by dg dt

+ kg =

G lOOV'

where k is a constant, G is the rate at which glucose is admitted, and V is the volume of blood in the body. Solve the differential equation and discuss the results.

33

1.6. Exercises

7. A chemical substance A changes into substance B at a rate a times the amount of A present. Substance B changes into C at a rate ß times the amount of B present. If initially only substance A is present and its amount is M, show that the amount of C present at time t is

M+M (

ße-at - ae-ßt) . a-ß

8. Two tanks, A and B, each ofvolume V, are filled with waterat timet= 0. For t > 0, volume v of solution containing mass m of solute ftows into tank A per second; mixture ftows from tank A to tank B at the same rate; and mixture ftows away from tank B at the same rate. The differential equations used to model this system are given by

v v das --+-as = -aA dt

V

V

'

where a A. B are the concentrations of solute in tanks A and B, respectively. Show that the mass of solute in tank B is given by

9. In an epidemic, the rate at which healthy people become infected is a times their number; the rates of recovery and death are, respectively, b and c times the number of infected people. If initially there are N healthy people and no sick people, find the number of deaths up to timet. ls this a realistic model? What other factors should be taken into account?

10.

(a) Determine the maximal interval of existence for each of the following initial value problems: (i)

x = x 4 , x(O) =

I;

(0) = 2 ; = x(x - 2), x(O) = 3.

x2-J .. ) • ( 11 X= ｾＬｘ＠

(iii)

x

(b) For what values of to and xo does the initial value problern

x = 2,JX,

x(to)

= xo,

have a unique solution?

Recommende d Textbooks [ 1] W. R. Derrick and S. I. Grossman, Elementary Differential Equations, Fourth ed., Addison-Wesley, Reading, MA, 1997.

34

1. Differential Equations

[2] K. Coombes, B. Hunt, R. Lipsman, J. Osbom, and G Stuck, Differential Equations with Maple, Second ed., John Wiley, New York, 1997. [3] R. Williams, lntroduction to Differential Equations and Dynamical Systems, McGraw-Hill, New York, 1997. [4] K. A. Stroud, Laplace Transforms, Stanley Thomes, Cheltenham, UK, 1989.

2 Linear Systems in the Plane

Aims and Objectives • To introduce the theory of planar autonomous linear differential equations. On completion of this chapter, the reader should be able to • classify critical points in the plane; • carry out simple linear transformations; • construct phase plane diagrams using isoclines, direction fields, and eigenvectors. Basic analytical methods for solving two-dimensional linear autonomous differential equations are reviewed and simple phase portraits are constructed in the plane.

2.1

Canonical Forms

Consider linear two-dimensional autonomous systems of the form dx (2.1)

.

dt = x = aux

+ ai2Y·

. dy dt = y = a21x

+ a22Y.

2. Linear Systems in the Plane

36

where the aij are constants. The system is linear as the terms in x, y, x, and alllinear. System (2.1) can be written in the equivalent matrix form as

y are

x=Ax,

(2.2) where x E 9l 2 and

A

= ( au

a21

=

(x(t), y(t)), can be Definition 1. Every solution of (2.1) and (2.2), say .. 1,2. cx, ß, and I.L arereal constants. matrix J 1 has two real distinct eigenvalues, matrix Jz has complex eigenvalues, and matrices J3 and J4 have repeated eigenvalues. The qualitative type of phase portrait is determined from each of these canonical forms. Nonsimple Canonical Systems. The linear system (2.2) is nonsimple ifthe matrix A is singular (i.e., det(A) = 0, and at least one of the eigenvalues is zero). The system then has critical points other than the origin. Example 1. Sketch a phase portrait of the system i

= x, y = 0.

Solution. The critical points are found by solving the equations i = y = 0, which has the solution x = 0. Thus there are an infinite nurober of critical points lying along the y axis. The direction field has gradient given by dy =

dx

ti = ｾ＠x =0

for x :/: 0. This implies that the direction field is horizontal for points not on the y axis. The direction vectors may be determined from the equation i = x since if x > 0, then i > 0, and the trajectories move from left to right; and if x < 0, then i < 0, and trajectories move from right to left. A phase portrait is plotted in Figure 2.1. Simple Canonical Systems. System (2.2) is simple if det(A) :/: 0, and the origin is then the only critical point. The critical points may be classified depending upon the type of eigenvalues. 2.1.1 Real Distinct Eigenvalues. Suppose that system (2.2) can be diagonalized to obtain i = AIX, y = A2Y· The solutions to this system are x(t) = CteAI 1 and y(t) = C2 eA.2t, where C 1 and Cz are constant. The solution curves may be found by solving the differential equation given by dy y >..zy -=-=-. dx i >.. 1x which is integrable. The solution curves are given by IYIA.1 = KlxiA.2. The type of phase portrait depends on the values of >..1 and >..z, as summarized below:

2. Linear Systems in the Plane

38 J""

..

·y ｾ＠

_",

-1 Ｍｾ＠

I ｘｾ＠

.

."

.i

. ｾｊ＠

....

Figure 2.1: Six trajectories and a vector field plot for Example 1. Note that there are an infinite number of critical points lying on the y axis.

• lf the eigenvalues are distinct, real and positive, then the critical point is called an unstable node. • If the eigenvalues are distinct, real and negative, then the critical point is called a stable node. • If one eigenvalue is positive and the other negative, then the critical point is called a saddle point or col. Possible phase portraits for these canonical systems along with vector fields superimposed are shown in Figure 2.2.

2.1.11 Complex Eigenvalues (l. = « :1: i{J). Consider a canonical system of the form (2.3)

x = ax

+ ßy,

y = -ßx + ay.

Convert to polar coordinates by making the transformations x = r cos (} and y = r sin (}. Then elementary calculus gives

rr =XX+ yy, System (2.3) becomes ;- = ar,

r 2Ö = xy- yx.

Ö = -ß.

The type of phase portrait depends on the values of a and ß:

39

2.1. Canonical Fonns

/'.?///11/13 /'/'.?//II/I //////II!

ｾ＠ HHif. \\\U\\

｜ｾ＠

\\\ \\\

\\\\

////////

__.__._.._._._.//

\\\

" ' " \\I ＧＢｾ＠ \\ .................... Ｌｾ｜＠

(a)

(b)

(c)

Figure 2.2: Possible phase portraits for canonical systems with two real distinct eigenvalues: (a) unstable node; (b) stable node; (c) saddle point (or col).

• If cx. > 0, then the critical point is called an unstable focus. • If cx.

= 0, then the critical point is called a center.

• If cx. < 0, then the critical point is called a stable focus. • If iJ > 0, then the trajectories spiral counterclockwise around the origin. • If iJ < 0, then the trajectories spiral clockwise around the origin. Phase portraits of the canonical systems with the vector fields superimposed are shown in Figure 2.3.

2.1.111 Repeated Real Eigenvalues. Suppose that the canonical matrices are of the form J3 or J4. The type of phase portrait is determined by the following: • If there are two linearly independent eigenvectors, then the critical point is called a singular node. • If there is one linearly independent eigenvector, then the critical point is called a degenerate node. Possible phase portraits with vector fields superimposed are shown in Figure 2.4. The classifications given in this section may be summarized using the trace and determinant of the matrix A as defined in system (2.2). If the eigenvalues are AJ,2, then the characteristic equation is given by (>..- >.. 1)(>..- >.. 2 ) >.. 2 - (>.. 1 +

=

40

2. Linear Systems in the Plane

(a)

(b)

(c)

Figure 2.3: Possible phase portraits for canonical systems with complex eigenvalues: (a) unstable focus; (b) stable focus; (c) center.

(b)

(a)

Figure 2.4: Possible phase portraits for canonical systems with repeated eigenvalues: (a) a stable singular node; (b) an unstable degenernte node.

A. 2)

+ A.t A.2 = )..2 -

trace(A)

AJ.2 =

+ det(A) = 0. Therefore,

trace(A) ±

./(trace(A) )2 - 4 det(A)

The summary is depicted in Figure 2.5.

2

.

2.2. Eigenvectors Defining Stable and Unstable Manifolds

s\ S N

41

DetA

Stable Focus

UnstabJe Focus Unstable Node

Stable Node Saddl point

TraceA

SSN - Stable Singular Node SDN - Stable Degenerate Node USN - Unstable Singular Node UDN - Unstable Degenerate Node

Figure 2.5: Classification of critical points for system (2.2). The parabola has equation T 2 - 4D = 0, where D = det(A) and T = trace(A).

2.2

Eigenvectors Defining Stahle and Unstable Manifolds

Consider Figure 2.5. Apart from the region T2 - 4D > 0, where the trajectories spiral, the phase portraits of the canonical forms of (2.2) all contain straight line trajectories that remain on the coordinate axes forever and exhibit exponential growth or decay along it. These special trajectories are determined by the eigenvectors of the matrix A and are called the manifolds. If the trajectories move towards the critical point at the origin as t --+ oo along the axis, then there is exponential decay and the axis is called a stable manifold. If trajectories move away from the critical point as t --+ oo, then the axis is called an unstable manifold. In the general case, the manifolds do not lie along the axes. Suppose that a trajectory is of the form x(t) = exp(i..t)e, where e :f: 0 is a vector and A. is a constant. This trajectory satisfies equation (2.2) since it is a solution curve. Therefore, substituting into (2.2), A.exp(AI)e = exp(A.t)Ae or A.e = Ae. From elementary linear algebra, if there exists a nonzero column vector e satisfying this equation, then A. is called an eigenvalue of A and e is called an eigenvector of A corresponding to the eigenvalue A.. If A. is negative, then the corresponding eigenvector gives the direction of the stable manifold, and if A. is positive, then the eigenvector gives the direction of the unstable manifold.

42

2. Linear Systems in the Plane

When >.. 1 =f. A.z, it is known from elementary linear algebrathat the eigenvectors e1 and ez, corresponding to the eigenvalues >.. 1 and A.z, are Iinearly independent. Therefore, the generat solution to the differential equations given by (2.1) is given by x(t) = C1 exp(A.1t)e1 + Cz exp(A.zt)ez, where C1 and C2 are constants. In fact, for any given initial condition, this solution is unique by the existence and uniqueness theorem. Consider the following two simple examples that illustrate these ideas. Example 2. Determine the stable and unstable manifolds for the linear system

x=2x+y.

y=x+2y.

Solution. The system can be written as x = Ax, where

A=(i ｾＩﾷ＠

The characteristic equation for matrix A is given by det(A - Al) = 0, or in this case, 1

2-}., 1

• I 2- >.. = O.

Therefore, the characteristic equation is >.. 2 - 4A. + 3 = 0, which has roots A.1 = 1 and A.z 3. Since both eigenvalues are real and positive, the critical point at the origin is an unstable node. The manifolds are determined from the eigenvectors corresponding to these eigenvalues. The eigenvector for A.1 is (1, -l)r and the eigenvector for A.z is (1, l)r. The manifolds are shown in Figure 2.6.

=

For the sake of completeness, the general solution in this case is given by x(t) = C1 exp(t)(l, -l)r

+ Czexp(3t)(l, l)r.

X

Figure 2.6: The unstable manifolds for Example 2.

2.3. Phase Portraits of Linear Systems in tbe Plane

43

Example 3. Detennine the stable and unstable manifolds for the linear system

i= (

］ｾ＠

ｾ＠

)x.

Solution. The characteristic equation for matrix A is given by 1

-3- A 4 I -2 3- A =

o.

Therefore, the characteristic equation is A2 - 1 = 0, which has roots AI = 1 and .A.2 = -1. Since one eigenvalue is real and positive and the other is real and negative, the critical point at the origin is a saddle point. The manifolds are derived from the eigenvectors corresponding to these eigenvalues. The eigenvector for AI is (1, l)T and the eigenvector for .A. 2 is (2, 1)T. The manifolds are shown in Figure 2.7.

Figure 2.7: The stable and unstable manifolds for Example 3. The trajectories lying on the stable manifold tend to the origin as t __. oo but never reach it. For the sake of completeness, the generat solution in this case is given by x(t) =CI exp(t)(1, l)T + C2 exp( -t)(2, l)T. A phase portrait for Example 2 is. plotted in the next section.

Notation. The stable and unstable manifolds of linear systems will be denoted by Es and Eu, respectively. Center manifolds (where the eigenvalues have zeroreal part) will be discussed in Chapter 8.

2.3

Phase Portraits of Linear Systems in the Plane

Definition 5. Two systems of first-order autonomous differential equations are said tobe qualitatively (or topologically) equivalent if there exists an invertible mapping that maps one phase portrait onto the other while preserving the orientation of the trajectories.

44

2. Linear Systems in the Plane

Phase portraits can be constructed using isoclines, vector fields, and eigenvectors (for real eigenvalues). Example 4. Consider the system

2I ) X.

( 2I . X=

Find (a) the eigenvalues and corresponding eigenvectors of A; (b) a nonsingular matrix P suchthat J = p-i AP is diagonal; (c) new coordinates (u, v) suchthat substituting x = x(u, v), y = y(u, v), converts the linear dynamical system

x=

2x

+ y,

y = x + 2y

into

u=

AiU,

v=

A2V

for suitable Ai and A2; (d) sketch phase portraits for these qualitativly equivalent systems. Solutions. The origin is a unique critical point. (a) From Example 2, the eigenvalues and corresponding eigenvectors are given by Ai = 1, (1, -l)r and A2 = 3, (1, l)r; the critical point is an unstable node. (b) Using linear algebra, the columns of matrix P are these eigenvectors and so

p

=(

I ) I

I -1

and J = p-iAP = (

ｾ＠ ｾＩＮ＠

(c) Take the linear transformation x = Pu. To obtain the system ü = u, 3v. (d) Consider the isoclines: In the xy plane, the flow is horizontal on the line where y = 0 and hence on the line = -x/2. On this line, = 3xf2; thus > 0 if x > 0 and x < 0 if x < 0. The flow is vertical on the line y = -2x. On this line, y < 0 if x > 0 and y > 0 if x < 0. Vector fields: The directions of the vector fields can be determined from and y at points (x, y) in the plane. Consider the slope of the trajectories: lf x + 2y > 0 and 2x + y > 0, then < 0; if x + 2y > 0 and 2x + y < 0, > 0; if x + 2y < 0 and 2x + y > 0, then

v=

x

y

**

*

x

x

*

> 0. < 0; and if x + 2y < 0 and 2x + y < 0, then Manifolds: From the eigenvectors, both manifolds are unstable; one passes through (0, 0) and (1, I) and the other through (0, 0) and (1, -1). Putting alloftbis informationtagether gives the phaseportraitinFigure 2.8(a). The canonical phase portrait is shown in Figure 2.8(b).

then

2.3. Phase Portraits of Linear Systems in the Plane

(a)

45

(b)

Figure 2.8: Qualitatively equivalent phase portraits for Example 4.

Example 5. Sketch a phase portrait for the system X = - X - y,

y =X -

y.

Solution. The origin is the only critical point. The characteristic equation is given by lA- All = )..2 + 2).. + 2 = 0, which has complex solutions Ä1,2 = -1 ± i. The critical point at the origin is a stable focus. Consider the isoclines: In the xy plane, the ftow is horizontal on the line where y = 0 and hence on the line y = x. On this line, x = -2x; thus x < 0 if x > 0 and x > 0 if x < 0. The ftow is vertical on the line where x = 0 and hence on the line y = - x. On this line, y < 0 if x > 0 and y > 0 if x < 0. Vector fields: The directions of the vector fields can be determined from .X and y at points (x, y) in the plane. Consider the slope of the trajectories: lf y > x and y > - x, then > 0; if y > x and y < - x, then < 0; if y < x and y > - x, then < 0; and if y < x and y < -x, then > 0. Manifolds: The eigenvectors are complex and there are no real manifolds. Converting to polar coordinates gives r = -r, iJ = 1. Putting all of this infonnation together gives the phase portrait in Figure 2.9.

* *

* *

46

2. Linear Systems in the Plane

Figure 2.9: Some trajectories for Example 5. The critical point is a stable focus.

Example 6. Sketch a phase portrait for the system

x = -2x, y = -4x- 2y. Solution. The origin is the only critical point. The characteristic equation is given by

lA -All= >.. 2 -4>..+4 = 0,

which has repeated roots >..1.2 = -2. Consider the isoclines: In the x y plane, the flow is horizontal on the line where y = 0 and hence on the line y = - 2x. Trajectories that start on the x axis remain there. Vector .fields: The directions of the vector fields can be determined from and y at points (x, y) in the plane. at each Consider the slope of the trajectories: The slopes are given by point (x, y) in the plane. Manifolds: There is one linearly independent eigenvector, (0, I) T. Therefore, the critical point is a stable degenerate node. The stable manifold Es is the y axis. Putting all of this tagether gives the phase portrait in Figure 2.1 0.

*

x

Phase portraits of nonlinear planar autonomaus systems will be considered in the next chapter, where stable and unstable manifolds do not necessarily lie on straight lines. However, all is not lost as the manifolds for certain critical points are tangent to the eigenvectors of the linearized system at that point. Manifolds in three-dimensional systems will be discussed in Chapter 8.

2.4. Maple Commands

47

Figure 2.10: Sorne trajectories for Exarnple 6. The critical point is a stable degenernte node.

2.4 Maple Commands The Maple cornrnands below rnay be edited to produce solutions and diagrarns for all exarnples and exercises in Chapter 2. >with(linal g) :with(DEto ols):with(p lots): > A:=matrix{ [[-3,4], [-2,3))); [ -3 A := [

4)

[-2

3)

)

> eigenvals (Al; 1, -1

> # Determine the stable and unstable manifolds. > eigenvects (Al ; [1, 1,

{[1, -1)}],

[3, 1, {[1, 1)})

> P:=matrix( [ [1,1), [-1,1) I); [1

p := [ [ -1

1)

l 1)

48

2. Linear Systems in the Plane

> J:=evalm(PA(-l)&*A&*P); [1 J

:=

0)

I

I

[0

3)

> deqnl:=diff(x(t),t)=2*x(t)+y(t),diff(y(t),t)=x(t)+2*y(t); d

d

deqnl :=

x(t) dt

2 x(t) + y(t), -- y(t) dt

x(t) + 2 y(t)

> iniset:={seq(seq([O,i,j],i=-2 .. 2),j=-2 .. 2)}: > > > >

# Plot a phase portrait for the linear system DEplot({deqnl}, [x(t),y(t)),-3 .. 3,iniset,stepsize=O.l,x=-3 .. 3, y=-3 .. 3,arrows=NONE,linecolor=blue,arrows=SLIM); # See Figure 2.8(a).

> # The canonical form, after diagonalization. > deqn2:=diff(x(t),t)=x(t),diff(y(t),t)=3y(t); d

d

deqn2 :=

x(t) = x(t), dt

y(t)

3y(t)

dt

DEplot({deqn2}, [x(t),y(t)),-3 .. 3,iniset,stepsize=O.l,x=-3 .. 3, y=-3 .. 3,arrows=NONE,linecolor=blue,arrows=SLIM); >#See Figure 2.8(b).

> >

2.5

Exercises

I. Sketch phase portraits for the following linear systems:

y = x + 2y; X =X+ 2y, y = 0.

(a) i = 0, (b)

2. Find the eigenvalues and eigenvectors of the matrix

-7 B = ( 2 Sketch a phase portrait for the system ical form.

6 ) -6 .

x= Bx and its corresponding canon-

3. Carry out the same procedures as in question 2 for the system

i = -4x - 8y,

y=

2y.

49

2.5. Exercises 4. Sketch phase portraits for each of the following linear systems: (a) (b) (c) (d)

x=

+ 4y, y = 4x - 3y; x = 3x + y, y = -x + 3y; x = y, y = -x - 2y; X= X - y, y = y- X. 3x

5. A very simple mechanical oscillator can be modeled using the second-order differential equation d 2x - 2

dt

dx

+ J.L+ 25x = dt

0,

where x measures displacement from equilibrium. (a) Rewrite this equation as a linear first-order system by setting x =

y.

(b) Sketch phase portraits when (i) J.L = -8;

(ii) J.L = 0; (iii) J.L = 8;

(iv) J.L = 26. (c) Describe the dynamical behavior in each case given that x(O) = 1 and x(O) = o. Plot the corresponding solutions in the tx plane.

Recommended Textbooks [1] J. Cronin, Differential Equations: lntroduction and Qualitative Theory, Second ed., Marcel Dekker, New York, 1994. [2] D. K. Arrowsmith and C. M. Place, Dynamical Systems, Chapman and Hall, London, 1992. [3] D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Second ed., Oxford University Press, Oxford, UK, 1990.

3 Nonlinear Systems in the Plane

Aims and Objectives • To extend the theory of linear systems to that of nonlinear systems. On completion of this chapter, the reader should be able to • find and classify all critical points in the plane; • construct phase portraits for nonlinear systems in the plane; • apply the theory to simple modeling problems. The method of linearization is introduced and both hyperbolic and nonhyperbolic critical points are defined. Phase portraits are constructed using Hartman Theorem. The linearization technique used here is based on a linear stability analysis (also used in later chapters) that was introduced in Chapter 1.

s

3.1

Linearization and Hartman's Theorem

Suppose that the nonlinear autonomaus system

x= (3.1)

y=

P(x, y), Q(x, y)

has a critical point at (u, v), where P and Q are at least quadratic in x and y. Take a linear transformation that moves the critical point to the origin. Let X = x - u

52

3. Nonlinear Systems in the Plane

and Y

=y-

v. Then system (3.1) becomes

aPI + u, Y + v) = P(u, v) +X-

+ YaPI

+ R(X, Y),

= Q(X + u, Y + v) = Q(u, v) + XöQI

+ YöQI

+ S(X, Y)

. P(X X=

ax x=u,y=v

öy x=u.y=v

(3.2)

Y.

ax x=u.y=v

ay x=u,y=v

after a Taylor series expansion. The nonlinear terms R and S satisfy the conditions ｾ＠ --+ 0 and ｾ＠ --+ 0 as r = J x2 + y2 --+ 0. The functions R and S are said to be "big Oh of r 2," or in mathematical notation, R = 0(r 2) and S = 0(r 2 ). Q(u, v) Discardthenonlineartermsins ystem(3.2)andnotethatP(u, v) 0 since (u, v) is a critical point of system (3.1 ). The linearized system is then of the form

=

(3.3)

aPI . XX= öx x=u,y=v

+ YaPI

öQI . X Y=

+ YöQI

öx x=u.y=v

and the Jacobian matrix is given by

J(u, v)

ｾ＠ ｾ＠ (

ay x=u,y=v

=

,

ay x=u,y=v

#)

x=•.Y='

Definition 1. A critical point is called hyperbolic if the real part of the eigenvalues ofthe Jacobian matrix J(u, v) are nonzero. Ifthe real part of either ofthe eigenvalues of the Jacobian are equal to zero, then the critical point is called nonhyperbolic. Hartman's Theorem. Suppose that (u, v) is a hyperbolic critical point ofsystem (3.1 ). Then there is a neighborhood ofthis critical point on which the phase portrait for the nonlinear system resembles that of the linearized system (3.3). In other

words, there is a curvilinear continuous change of coordinates taking one phase portrait to the other, and in a small region around the critical point, the portraits are qualitativly equivalent. A proof to this theorem may be found in Hartman's book [3]. Note that the stable and unstable manifolds of the nonlinear system will be tangent to the manifolds of the linearized system near the relevant critical point. These trajectories diverge as one moves away from the critical point; this is illustrated in Examples l and 2. Notation. Stahle and unstable manifolds of a nonlinear system are labeled Ws and Wu, respectively.

53

3.2. Constructing Phase Plane Diagrams

Hartman's Theorem implies that Ws and Wu are tangent to Es and Eu at the relevant critical point. If any of the critical points are nonhyperbolic, then other methods have to be used to sketch a phase portrait, and numerical solvers may be required.

3.2

Constructing Phase Plane Diagrams

The method for plotting phase portraits for nonlinear planar systems having hyperbolic critical points may be broken down into three distinct steps: • Locate all of the critical points. • Linearize and classify each critical point according to Hartman's Theorem. • Determine the isoclines and use ｾ＠

to obtain slopes of trajectories.

The method can be illustrated with some simple examples. Examples 4-6 illustrate possible approaches when a critical point is not hyperbolic.

Example 1. Sketch a phase portrait for the nonlinear system

X =X, Solution. Locate the critical points by solving the equations x = y = 0. Hence x = 0 if x = 0 and y = 0 if x 2 + y 2 = l. If x = 0, then y = 0 if y 2 = I, which has solutions y = 1 and y = -l. Therefore, there are two critical points, (0, 1) and (0, -I). Linearize by finding the Jacobian matrix; hence iJP

1=

(

ｾ＠

ax iJx

ｾＩﾷ＠ ｾＩ］ＨｩＮ＠

Linearize at each critical point; hence

The matrix is in diagonal form; there are two distinct positive eigenvalues, and hence the critical point is an unstable node. For the other critical point, l(0,-1)

= (

ｾ＠ ｾＲ＠

) .

There is one positive and one negative eigenvalue, and so this critical point is a saddle point or col.

54

3. Nonlinear Systems in the Plane

Note that the matrices 1(0,1) and J(o.-1) are in diagonal form. The eigenvectors for both critical points are (1, 0) T and (0, I) T. Thus in a small neighborhood around each critical point, the stable and unstable manifolds are tangent to the lines generated by the eigenvectors through each critical point. Therefore, near each critical point the manifolds are horizontal and vertical. Note that the manifolds of the nonlinear system Ws and Wu need not be straight lines but are tangent to Es and Eu at the relevant critical point. Consider the isoclines. Now i = 0 on x = 0, and on this line, y = y 2 - 1. Thus if IYI < 1, then y < 0, and if IYI > 1, then y > 0. Also, y = 0 on the circle x 2 + y 2 = 1, and on this curve, i = x. Thus if x > 0, then i > 0, and if x < 0, then i < 0. The slope of the trajectories is given by dy x 2 + y2 - 1 = - -x - dx

Putting all of this together gives a phase portrait, as depicted in Figure 3.1.

Figure 3.1: A phase portrait for Examp1e I. The stable and unstable manifolds (Ws, Wu) are tangent to horizontal or verticallines (Es, Eu) in a small neighborhood of each critical point. Example 2. Sketch a phase portrait for the nonlinear system

i =y,

y=

x(l - x 2)

+ y.

55

3.2. Constructing Phase Plane Diagrams

Solution. Locate the critical points by solving the equations x = y = 0. Hence x = 0 if y = 0 and y = 0 if x(l - x 2) + y = 0. lf y = 0, then y = 0 if x(l- x2) = 0, which has solutions x = 0, x = 1, and x = -1. Therefore, there are three critical points, {0, 0), (1, 0), and ( -1, 0). Linearize by finding the Jacobian matrix; hence

aP ax ( 1= ｾ＠

ax

Linearize at each critical point; hence 1(0,0}

= (

ｾ＠

! ).

The eigenvalues are A.t = 1 + .J5

and

2

A2 =

1-

.J5 .

2

The corresponding eigenvectors are (1 A. 1) T and (1 A.2) T. Thus the critical point at the origin is a saddle point or col. For the other critical points,

= 1(-1,0} = ( ｾ Ｒ＠

1(t,O)

The eigenvalues are

).. =

) .

1 ±iJ1 2 '

and so both critical points are unstable foci. Consider the isoclines. Now x = 0 on y = 0, and on this line, y = x(l- x 2 ). Thus if 0 < x < 1, then y > 0; if x > 1, then y < 0; if -1 < x < 0, then y < 0, and if x < -1, then y > 0. Also, y = 0 on the curve y = x- x 3 , and on this curve, x = y. Thus if y > 0, then x > 0, and if y < 0, then x < 0. The slope of the trajectories is given by

*

Note that on x = 0 and x = ±1, portrait, as depicted in Figure 3.2.

= 1. Putting alloftbis together gives a phase

Example 3. Plot a phase portrait for the system

ｸ］ＨｴＭｾｹＩＬ＠

y=

y

(x- ｉＭｾＩＮ＠

3. Nonlinear Systems in the Plane

56

Figure 3.2: A phase portrait for Example 2. Note that in a small neighborhood of the origin, the unstable manifold (Wu) is tangent to the line Eu given by y = A1x, and the stable manifold (Ws) is tangent to the line Es given by y = A2X; the equations of both lines are derived from the respective eigenvectors.

Solution. Locate the critical points by solving the equations x = y = 0. Hence x = 0 if either x = 0 or y = I - ! . Suppose that x = 0. Then y = 0 if y(-1- ｾＩ＠ = 0, which has solutions y = 0 or y = -2. Suppose that y = 1 -1· Then y = 0 if either 1 - ! = 0 or 1 - ! = 2x - 2, which has solutions x = 2 ｾＩＮ＠ or x = ｾＭ Thus there are four critical points at (0, 0), (2, 0), (0, -2), and ＨｾＮ＠ Notice that x = 0 when x = 0, which means that the ftow is vertical on the y axis. Similarly, y = 0 when y = 0, and the ftow is horizontal along the x axis. In this case, the axes are invariant. Linearize by finding the Jacobian matrix; hence öP

J

=

(

äX

!!Jl

ax

öP ) _ ( 1- X - y 3y

!!Jl ay

-

y

-X

x-1-y

)

·

Linearize around each of the critical points and apply Hartman's Theorem. Consider the critical point at (0, 0). The eigenvalues are A = ± 1 and the critical point is a sadd1e point (or col). Next, consider the critical point at (2,0); now the eigenvalues are AJ = 1 and A2 = -1. The corresponding eigenvectors are (-1, l)r and (l,O)r, respectively. This critical point is also a saddle point (or col). Consider the critical point at (0, -2). Now the eigenvalues are AJ = 3 and A2 = I; the corresponding eigenvectors are (1, -l)r and (0, l)r, respectively.

57

3.2. Constructing Phase Plane Diagrams

The critical point at (0, -2) is therefore an unstable node. Finally, consider the ｾＩＮ＠ The eigenvalues in this case are critical point at ＨｾＮ＠

-2 ± ;.JIT 'A=----

5

and the critical point is a stable focus. There is no need to find the eigenvectors that will be complex in this case. and y 0 on Consider the isoclines. Now = 0 on x = 0 or on = 1 y = 0 or on y = 2x - 2. The directions of the flow can be found by considering y and on these curves. The slope of the trajectories is given by

y

x

!,

=

x

dy

y

(x- 1- ｾＩ＠

dx = x ( 1 -

! - y)"

A phase portrait indicating the stable and unstable manifolds of the critical points is shown in Figure 3.3.

ｾｉ＠

II

1111/l ///II II/

Figure 3.3: A phase portrait for Example 3. The axes are invariant.

Example 4. Sketch a phase portrait for the nonlinear system

58

3. Nonlinear Systems in the Plane

Solution. Locate the critical points by solving the equations .i = y = 0. Therefore, = 0 if y = 0 and y = 0 if x = 0. Thus the origin is the only critical point.

.i

Attempt to linearize by finding the Jacobian matrix; hence IJP

J =

(

ax

!Cl

ax

Linearize at the origin to obtain

Jco.o> = (

ｾ＠ ｾ＠

) ·

The eigenvalues are both zero and so the origin is a nonhyberbolic critical point. To sketch a phase portrait, solve the differential equation

dy dx

y

X

=I= y2'

This differential equation was solved in Chapter 1, and the solution curves were given in Figure 1.2. Consider the isoclines. Now .i 0 on y 0, and on this line, y x. Thus if x > 0, then y > 0, and if x < 0, then y < 0. Also, y 0 on x 0, and on this Iine, i = y 2• Thus i > 0 for all y. The slope ofthe trajectories is given by ｾ＠ = ｾﾷ＠ Putting all of this together gives a phase portrait, as depicted in Figure 3.4.

=

=

_

=

=

=

_._............

ｾＭﾷＫ｟ＮＯ＠

_ . / / // / /

ｾＭＯ＠

Figure 3.4: A phase portrait for Example 4 that has a nonhyperbolic critical point at the origin. There is a cusp at the origin.

3.2. Constructing Phase Plane Diagrams

59

Example 5. A simple model for the spread of an epidemic in a city is given by

S=

i = "CSI -rl,

-•SI,

where S (t) and I (t) represent the numbers of susceptible and infected individuals scaled by one thousand, respectively; "C is a constant measuring how quickly the disease is transmitted; r measures the rate of recovery (assume that those who recover become immune); and t is measured in days. Determine a value for S at which the infected population is a maximum. Given that "C = 0.003 and r = 0.5, sketch a phase portrait showing three trajectories whose initial points are at (1000, 1), (700, 1), and (500, 1). Give a physical interpretation in each case. Solution. The maximum number of infected individuals occurs when ｾ＠ Now dl i "CS-r dS = = -•S

= 0.

S

Therefore, ｾ＠ = 0 when S = f. The number f is called the threshold value. The critical points for this system are found by solving the equations S = i = 0. Therefore, there are an infinite number of critical points lying along the x axis. A phase portrait is plotted in Figure 3.5.

Figure 3.5: A phase portrait showing three trajectories for Example 5. The axes are scaled by toJ in each case. Trajectories are only plotted in the first quadrant since populations cannot be negative. In each case, the population of susceptibles decreases to a constant value and the population of infected individuals increases and then decreases to zero. Note that in each case, the maximum number of infected individuals occurs at s = ｾ＠ ｾ＠ 167. 000.

60

3. Nonlinear Systems in the Plane

Example 6. Chemical kinetics involving the derivation of one differential equation were introduced in Chapter I. This example will consider a system of two differential equations. Consider the isothermal chemical reaction

A+B ｾｃＮ＠

in which one molecule of A combines with one molecule of B to form one molecule of C. In the reverse reaction, one molecule of C returns to A + B. Suppose that the rate of the forward reaction is k 1 and the rate of the backward reaction is kr. Let the concentrations of A, B, and C be a, b, and c, respectively. Assurne that the concentration of A is much larger than the concentrations of B and C and can therefore be thought of as constant. From the law of mass action, the equations for the kinetics of b and c are

Find the critical points and sketch a typical trajectory for this system. Interpret the results in physical terms.

c

Solution. The critical points are found by determining where b = = 0. Clearly, The slope of the there are an infinite number of critical points along the line ｾﾷ＠ trajectories is given by

dc

c

-=-:-=-1. b db

c

c

then b > 0 and < 0. then b < 0 and > 0. Similarly, if c > ｾＬ＠ If c < ｾＬ＠ Two typical solution curves are plotted in Figure 3.6. Thus the final concentrations of B and C depend on the initial concentrations of these chemicals. Two trajectories starting from the initial points at (bo, 0) and (bo, co) are plotted in Figure 3.6. Note that the chemical reaction obeys the law of conservation of mass. This explains why the trajectories lie along the Iines b + c = constant. Example 7. Suppose that H is a population of healthy rabbits and I is the Subpopulation of infected rabbits that never recover once infected, both measured in millions. The following differential equations can be used to model the dynamics of the system:

H = (b -d)H- 81, i = r:l(H - / ) - (8 +d)l, where b is the birth rate, d is the natural death rate, 8 is the rate of death of the diseased rabbits, and r: is the rate at which the disease is transmitted. Given that b = 4, d = 1, 8 = 6, and r: = 1 and given an initia1 population of (Ho, /o) = (2, 2), plot a phase portrait and explain what happens to the rabbits in real-world terms.

61

3.3. Maple Commands \\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\ ｜ｾ＠

｜ｾ＠

c \\\\\\\\\ \\\\\\\\\ ｜ｾ＠ ｜ｾ＠

｜ｾ＠

｜ｾ＠ ｜ｾ＠

\\\\\\\\\\\\\\ \\\\\\\\\\\\ \\\\\\\\\\ \\\\\\\\

\\\\\\

\\\\ \\

Figure 3.6: Two solution curves for the chemical kinetic equation in Example 6, where a is assumed to be constant. The dotted line represents the critical points lying on the line c = ｾ｢Ｎ＠ Solution. There are two critical points in the first quadrant at 0 = (0, 0) and P = (14, 7). The Jacobian matrix is given by l=

( (b- d) ri

-8 ) rH-2r/-(8+b) ·

The critical point at the origin is a col with eigenvalues and corresponding eigenvectors given by >..1 = 3,(1,0)r and >..2 = -7,(3,5)r. The critical point at P = (14, 7) has eigenvalues >.. = -2 ± ;ffi and is therefore a stable focus. A phase portrait is plotted in Figure 3.7. Either the population of rabbits stabilizes to the values at P or they become extinct, depending on the initial populations. For example, plot a solution curve for the trajectory starting at (Ho, /o) = (7, 14). Models of interacting species will be considered in Chapter 4.

3.3

Maple Commands

The Maple commands below and in Chapter 2 may be edited to produce solutions and diagrams for all of the examples and exercises appearing in Chapter 3. >?mtaylor >readlib(mtaylor): >mtaylor(P(X+u,Y+v), [X,Y),3);

3. Nonlinear Systems in the Plane

62

f']llllllllllllm 28

26

24 22

2o 18

I I I I l ll II II I I/III i I I I / l/1 J/ II I/-\ I II I I I 11111111 I/-\\ ll/lJ/11111111 ｾ｜＠ II /1111 II /II I -\

I I I II I I I I I II. -\ \ \\ I I I I I II/ I I ＱＮｾＭ

HHuHｾｉＢ｜＠ 'j jl 1) ) f I d ｾ＠ uu1V'111/1/1/17/1 Ｚｾ＠ 111/11

16

14

6

ＮＬｲ［ｾＭＺ＠

4 2

111111"'11!1/l!l II/ /II ｬ｜ｾＯＱｨ＠ ｬ｜ｾＭＯ＠

/////

............-......pealthy

2 4 6 810 14 18 22 26 30 34

Figure 3.7: A trajectory starting from the initial point (2, 2). The population stabilizes to I4 million healthy rabbits and 7 million infected rabbits.

P(u, v)

I

+ Dt (P)(u, v)X + D2(P)(u, v)Y + 2Dt,l (P)(u, v)X 2 1

+ D1,2(P)(u, v)XY + 2n2.2(P)(u, v)Y 2 >with(DEtools): >sys:=diff(x(t),t)=x(t),d iff(y(t),t)=(x(t))'2+(y(t)) '2-l: >iniset:=seq(seq( (0, i, j 1. i=-2 .. 2), j=-2 .. 2): >DEplot({sys}, [x(t) ,y(t) I ,t=-3 .. 3, {iniset, [0,0.1,-11, [0,-0.1,-11}, >stepsize=0.1,x=-3 .. 3,y=-3 .. 3,color=b1ack, >linecolor=b1ack); ># See Figure 3.1. >sys3:=diff(x(t),t)=x(t)* (1-0.5*x(t)-y(t)),diff(y(t ),t) >=y(t)*(-1-0.S*y(t)+x(t)); >DEplot ( [ sys31 , [x ( t) , y ( t) I , t=O .. 50, [ [ 0, 3, 21, [ 0, 0. 5, 31 , [ 0, 1. 8, 0. 0511 , >stepsize=0.1,x=O .. 3,y=0 .. 3,color=black,linecolor=bl ack); >#See Figure 3.3. >ODE:=diff(y(x),x)=(y(x)) '2/x'2: >dsolve (ODE); X

y(x)

3.4

= I+ Cx

Exercises

1. Plot phase portraits for the following systems:

63

3.4. Exercises

i (b) i (a)

= y, y = x - y + x 3 ; = -2x- y + 2, y = xy.

2. Plot phase plane diagrams for the following systems:

i = x 2 - y 2, y = xy- 1; (b) i = 2- x- y 2, y = -y(x 2 + y 2 - 3x + 1). (a)

3. Construct a nonlinear system that has four critical points: two cols, one stable focus, and one unstable focus. 4. Plot phase portraits for the following nonlinear systems:

i = y2, y = x 2 ; (b) i = x 2 , y = y 2 ; (a)

(c)i=y,y=x 3 .

5. Consider the system

i=x, Sketch phase portraits when

J.L




0.

6. A nonlinear capacitor-resistor electrical circuit can be modeled using the differential equations

i

= y, y = -x + x 3 -

(ao

+ x)y,

where ao is a nonzero constant and x(t) represents the current in the circuit at timet. Sketch phase portraits when ao > 0 and ao < 0 and give a physical interpretation of the results. 7. An age-dependent population can be modeled by the differential equations

P = ß + p(a- bp),

ß=

ß(c + (a- bp)),

where p is the population, ß is the birth rate, and a, b, and c areallpositive constants. Find the critical pointsoftbis system and determine the Iang-term solution. 8. The power, say P, generated by a water wheel of velocity V can be modeled by the system .

P = -a.P + PV,

•

2

V= 1- ßV- P ,

where a and ß are both positive. Describe the qualitative behaviour of this system as a and ß vary and give physical interpretations of the results.

3. Nonlinear Systems in the Plane

64

9. A very simple model for the economy is given by

i =I -

K S,

S= I -

C S - Go,

where I represents income, S is the rate of spending, Go denotes govemment spending, and C and K are positive constants. (a) Plot possible solution curves when C = 1 and interpret the solutions in economic terms. What happens when C :f: 11 (b) Plot the solution curves when K = 4, C = 2, Go= 4, I(O) = 15, and S(O) = 5. What happens for other initial conditions? 10. Given that

and X=

11*

-2-,

ｾ＠

prove that X

= X(l +X + y), y = y(2 +X -

y).

Plot a phase portrait in the xy plane.

Recommended Textbooks [1] B. WestS. Strogatz, J. M. McDill, J. Cantwell, and H. Hohn, Interactive Differential Equations, Version 2.0, Addison-Wesley, Reading, MA, 1997. [2] E. J. Kostelieh and D. Armbruster, Introductory Differential Equations, Addison-Wesley, 1997. [3] P. Hartman, Ordinary Differential Equations, John Wiley, New York, 1964.

4 Interacting Species

Aims and Objectives • To apply the theory of planar systems to modeling interacting species. On completion of this chapter, the reader should be able to • plot solution curves to modeling problems for planar systems; • interpret the results in terms of species behavior. The theory of planar ordinary differential equations is applied to the study of interacting species. The models are restricted in that only two species are considered and external factors such as pollution, environment, refuge, age classes, and other species interactions, for example, are ignored. However, even these restricted systems give useful results. These simple models can be applied to species living in our oceans and to both animaland insect populations on land. Note that the continuous differential equations used in this chapter are only relevant if the species populations under consideration are large, typically scaled by 1o4, 1os, or 106 in applications.

4.1

Competing Species

Suppose that there are two species in competition with one another in an environment where the common food supply is limited. For example, sea lions and penguins, red and grey squirrels, and ants and termites are alt species that fall into

4. Interacting Species

66

this category. There are two particular types of outcome that are often observed in the real world. In the first case, there is coexistence, in which the two species live in harmony. (In nature, this is the most likely outcome; otherwise, one of the species would be extinct.) In the second case, there is mutual exclusion, in which one of the species becomes extinct. (For example, American grey squirrels imported into the U.K. are causing the extinction of the smaller native red squirrels.) Both coexistence and mutual exclusion can be observed when plotting solution curves on a phase plane diagram. Consider the following generat model for two competing species. Example 1. Sketch possible phase plane diagrams for the following system:

x=

x(ß - 8x - yy), y = y(b- dy- cx),

(4.1)

where ß, 8, y, a, b, and c are all positive constants with x(t) and y(t)-both positive-representing the two species populations measured in tens or hundreds of thousands. Solution. The terms appearing in the right-hand sides of equation (4.1) have a physical meaning as follows: • The terms ßx- 8x 2 and by- dy 2 represent the usuallogistic growth of one species (Verhulst's equation). • Both species suffer as a result of competition over a limited food supply, hence the terms - y x y and -cx y in and y.

x

Construct a phase plane diagram in the usual way. Find the critical points, linearize around each one, determine the isoclines, and plot the phaseplane portrait. Locate the critical points by solving the equations x = y = 0. There are four critical points at

o = (0,0),

P

=

(o. ｾＩＮ＠

Q=

ＨｾＮｯＩ＠

and R

= Ｈｾ＠

］ｾＺＮ＠

ｾＺ＠

=!:).

Suppose that c, = yc- 8d, C2 = yb- ßd, and C3 = ßc- 8b. In order for the critical point to lie in the first quadrant, one of the following conditions must hold: Bither (i)

c,, C2, and C3 are all negative, or

(ii) C 1, C2, and C3 are all positive. Linearize by finding the Jacobian matrix. Therefore,

_ ( ß- 28x- yy 1 -cy

) -yx b- 2dy- cx ·

67

4.1. Competing Species Linearize at each critical point. Thus Jo = (

ｾ＠ ｾＩＮ＠

For the critical point at P, J

- (

P-

ß - ybfd -bcfd

0 ) -b ·

For the critical point at Q, lQ =

( -ß 0

-yß/8 ) b- ßc/8 ·

Finally, for the critical point at R,

I ( 8C2 J __ R- Cl CCJ

yC2 ) dC3 .

Consider case (i) first. The fixed points are all simple and it is not difficult to show that 0 is an unstable node, P and Q are cols, and for certain parameter values, R is a stable fixed point. A phase portrait is plotted in Figure 4.1 (a), where eight of an infinite number of solution curves are plotted. Bach trajectory is plotted numerically for both positive and negative time steps; in this way, critical points are easily identified in the phase plane. For the parameter values chosen here, the two species coexist and the populations stabilize to constant values after long time periods. The arrows in Figure 4.l(a) show the vector field plot and define the direction of the trajectories for system (4.1 ). The slope of each arrow is given by ｾ＠ at the point, and the direction of the arrows is determined from ｾ＠ and There is a stable node lying wholly in the first quadrant at R, and the nonzero populations x(t) and y(t) tend to this critical point with increasing time no matter what the initial populations are. The domain of stability for the critical point at R is therefore SR = {(x, y) E !R 2 : x > 0, y > 0}. Now consider case (ii). The fixed points are all simple, and it is not difficult to show that 0 is an unstable node, P and Q are stable or improper nodes, and R is a col. A phase portrait is shown in Figure 4.l(b), where nine of an infinite number of solution curves are plotted. Once more the trajectories are plotted for both positive and negative time iterations. In this case, one of the species becomes extinct. In Figure 4.1 (b ), the critical point lying wholly in the first quadrant is a saddle point (or col), which is unstable. The long-term behavior of the system is divided along the diagonal in the first quadrant. Trajectories starting to the right of the diagonal will tend to the critical point at Q = (2, 0}, which implies that species y becomes extinct. Trajectories starting to the left of the diagonal will tend to the critical point at P = {0, 2), which means that species x will become extinct.

*.

4. Interacting Species

68

(b)

(a)

Figure 4.1: (a) A possible phase portrait showing coexistence and (b) a possible phase portrait depicting mutual exclusion. Note that the axes are invariant in both cases. Numerically, the trajectories lying on the stable manifold of the saddle point in the first quadrant will tend towards the critical point at R. However, in the real world, populations cannot remain exactly on the stable manifold, and trajectories will be diverted from this critical point, leading to the extinction of one of the (0, 2) is given by species. The domain of stability for the critical point at P 2 for the critical stability of domain S p = {(X, y) E 9l : X > 0, y > 0, y > X}. The 2 : X > 0, y > 0, y < x}. point at Q = (2, 0) is given by SQ = {(x, y) E

=

m

4.2

Predator-Prey Models

Consider a two-species predator-prey model in which one species preys on another. Examples in the natural world include sharks and fish, lynx and snowshoe bares, and ladybirds and aphids. A very simple differential equation-first used by Valterra in 1928 [4, 5] and known as the Lotka-Volterra model-is given in Example 2.

Example 2. Sketch a phase portrait for the system

= x(a- cy), y = y(yx ＭｾＩＮ＠

i

(4.2)

where a, c, y, and ｾ＠ areallpositive constants, with x(t) and y(t) representing the scaled population of prey and predator, respectively, and t is measured in years.

4.2. Predator-Prey Models

69

Solution. The terms appearing in the right-hand sides of equation (4.2) have a physical meaning as follows: • The term ax represents the growth of the population of prey in the absence of any predators. This is obviously a crude model; the population of a species cannot increase forever. • The terms -cx y and +y x y represent species interaction. The population of prey suffer and predators gain from the interaction. • The term

-oy represents the extinction of predators in the absence of prey.

Attempt to construct a phase plane diagram in the usual way. Find the critical points, linearize around each one, determine the isoclines, and plot the phaseplane portrait. The critical points are found by solving the equations = y = 0. There are two critical points, one at 0 = (0, 0) and the other at P = ( %). Linearize to obtain

x

f,

_ ( a -cy

J-

YY

-cx ) + yx .

ｾ＠ -o

The critical point at the origin is a saddle point, and the stable and unstable manifolds lie along the axes. The stable manifold lies on the positive y axis and the unstable manifold lies on the x axis. The critical point at P is not hyperbolic, so Hartman's Theorem cannot be applied. System (4.2) has solution curves (the differential equation is separable) given by x 6 yae-yxe-cy = K, where K is a constant. These solution curves may be plotted in the phase plane. The isoclines are given by x = 0, y = %, where the flow is vertical, and y = 0, x = where the flow is horizontal. The vector fields are found by considering y, and ｾＮ＠ A phase portrait is shown in Figure 4.2. The population fluctuations can also be represented in the tx and ty planes. The graphs shown in Figure 4.3 show how the populations of predator and prey oscillate. Note that the oscillations are dependent on the initial conditions. In Figure 4.3, the period of both cycles is about 10 years. Different sets of initial conditions can give solutions with different amplitudes. For example, plot the solution curves in the tx and ty planes for the initial conditions x(O) = 1 and y(O) = 3. How can this system be interpreted in terms of species behavior? Consider the trajectory passing through the point (1, 3) in Figure 4.2. At this point, the ratio of predators to prey is high; as a results the population of predators drops. The ratio of predators to prey drops, and so the population of prey increases. Once there are lots of prey, the predator numbers will again start to increase. The resulting cyclic behavior is repeated over and over and is shown as the largest closed trajectory in Figure 4.2.

x,

f,

4. Interacting Species

70

Figure 4.2: A phase portrait for the Lotka-Volterra model.

4

4

3

3

10

(a)

20

t 30

40

50

(b)

Figure 4.3: (a) Periodic behavior of the prey and (b) periodic behavior of the predators for one set of initial conditions, namely x(O) = 1, y(O) = 2.

If small perturbations are introduced into system (4.2)-to modelother factors, for example-then the qualitative behavior changes. The periodic cycles can be destroyed by adding small terms into the right-hand sides of system (4.2). The system is said tobe structurally unstable (or not robust).

71

4.2. Predator-Prey Models

Many predator-prey interactions have been modeled in the natural world. For example, there are data dating back over 150 years for the populations of lynx and snowshoe bares frorn the Hudson Bay Cornpany in Canada. The data clearly show that the populations periodically rise and fall (with a period of about ten years) and that the rnaxirnurn and minimurn values (amplitudes) are relatively constant. This is not true for the Lotka-Valterra model. Differentinitial conditions can give solutions with different arnplitudes. In 1975, Holling and Tanner constructed a system of differential equations whose solutions have the same amplitudes in the long term, no matter what the initial populations. Two particular examples of the Holling-Tanner rnodel for predator-prey interactions are given in Example 3. The reader is encouraged to compare the terms (and their physical rneaning) appearing in the right-hand sides of the differential equations in Examples 1-3.

Example 3. Consider the specific Holling-Tanner model

.

X

=

X

(

.

(4.3)

1-

y=0.2y

x)

7 -

6xy

(7 + 7X) '

Ny) ' (1--;-

where N is a constant with x(t) :f= 0 and y(t) representing the populations of prey and predators, respectively. Sketch phase portraits when (i) N = 2.5 and (ii) N = 0.5.

Solution. The terms appearing in the right-hand sides of equation (4.3) have a physical meaning as follows: • The terrn x (1 predators.

;f)

represents the usual logistic growth in the absence of

• The terrn - (76_t.tx) represents the effect of predators subject to a maximum predation rate. denotes the predator growth rate when a maximum • The terrn 0.2y(l - ｾＩ＠ by x prey. supported is of x IN predators Construct a phase plane diagram in the usual way. Find the critical points, linearize around each one, deterrnine the isoclines, and plot a phase plane portrait. Consider case (i). The critical points are found by solving the equations i = y = 0. There are two critical points in the first quadrant, A = (5, 2) and B = (7, 0). The Jacobian rnatrices are given by JA= (

>..2

-1

0

-3/4) 1/5

and

JB

=(

-10/21 2125

-5/7) -1/5 .

The eigenvalues and eigenvectors of JA are given by A. 1 = -1; (1, O)T and = 1I 5; (- i, 1) T. Therefore, this critical point is a saddle point (or col) with the

4. Interacting Species

72

stable manifold lying along the x axis and the unstable manifold tangent to the line with slope in a small neighborhood around the critical point. The eigenvalues of Js are given by J.. ｾ＠ -0.338 ± 0.195i. Therefore, the critical point at Bis a stable focus. A phase portrait showing four trajectories and the vector field is shown in Figure 4.4(a).

-J

4

4

3

3

y 2

0

3 x4

2

5

6

7

(a)

(b)

Figure 4.4: (a) A phase portrait for system (4.3) when N = 2.5. (b) Intersection of the isoclines. The populations eventually settle down to constant values. If there are any natural disasters or diseases, for example, the populations would both decrease but eventually retum to stable values. This is, of course, assuming that neither species becomes extinct. There is no periodic behavior in this model. Consider case (ii). The critical points are found by solving the equations i = y = 0. There are two critical points in the first quadrant, A = ( 1, 2) and B = (7, 0). The Jacobian matrices are given by JA= (

-1

0

-3/4) l/5

and

2/7 Js = ( 2/5

-3/7 ) -1/5 .

The eigenvalues and eigenvectors of JA are given by A.t = -1; {1, O)T and >.. 2 = 1/5; 1)T. Therefore, this critica1 point is a saddle point (or col) with the stable manifold lying along the x axis and the unstable manifold tangent to the line with slope - near to the critical point.

with(linalg): >with(plots): >with(DEtoolsl: >

># See Example 1. >solve({x*(beta-delta*x-gam ma*y),y*(b-d*y-c*x)},{x,y} );

75

4.5. Exercises b) (ß ) (yb-ßd ßc-8b) (O, O), ( O, d ' 8' 0 ' yc- 8d' yc- 8d ＾ｊｾ］ｭ｡ｴｲｩｸＨ｛｢･Ｍ

gamma*b/d,Ol, [-b*c/d,-blll:

J = (

ß

-/!

-([

0 ) -b

>det (J);

>trace(J);

yb ß---b d

># The Lotka-Volterra model ＾ｳｹｾ］､ｩｦＨｸｴＩＬＪＱＭＰＮＳＺ＠

>DEplot( [sys], [x(t) ,y(t)) ,t=-10 .. 10, [ [x(0)=1,y(0)=1.2), >[x(0)=1,y(0)=2], [x(O)=l,y(0)=3)), >stepsize=0.1,titl e='Lotka--Volterr a model',color=black , >linecolor=black) ; ># See Figure 4.2. ># Plotting the isoclines .. 7,y=0 .. 4.5); ＾ｰｬｯｴＨｻＭｸｾＲＯＶＫＷＬＪｽ］Ｐ＠

>#See Figure 4.5(b).

4.5

Exercises

1. Plot a phase portrait for the cornpeting species rnodel . 2x-x 2 -xy, x=

y=

3y-

i- 2xy

and describe what happens in terms of species behavior. 2. Plot phase plane diagrarns for the following predator-prey systerns and interpret the solutions in terms of species behavior: (a) (b)

x = 2x x = 2x -

xy,

y=

x 2 - xy,

-3y + xy;

y = -y- y 2 + xy.

3. The differential equations used to rnodel a cornpeting species are given by

x = x(2- x- y), y = y(/.L- y- 11- 2x), where 11- is a constant. Describe the qualitative behavior of this systern as the pararneter 11- varies.

76

4. lnteracting Specles

4. Suppose that there are three species of insect, say X, Y, and Z. Give rough sketches to illustrate the possible ways in which these species can interact with one another. You should include the possibility of a species being cannibalistic. Three-dimensional systems will be discussed later.

5. The following three differential equations are used to model a combined predator-prey and competing species system:

x = x(aw -

a11x

+ a12Y -

a13z),

.Y = y(a20 - a21x - a22Y - a23z),

(4.4)

z=

z(a3o

+ a31x- a32Y- a33Z),

where aij are all positive constants. Give a physical interpretation for the terms appearing in the right-hand sides of these differential equations.

Recommended Reading [1] Y. Lenbury S. Rattanamongkonkul, N. Tumrasvin, and S. Amomsamankul, Predator-prey interaction coupled by parasitic infection: Limit cycles and chaotic behaviour, Math. Comput. Model., 30-9110 (1999), 131-146. [2] R. E. Kooij and A. Zegeling, Qualitative properlies of two-dimensional

predator-prey systems, Nonlinear Anal. Theory Meth. Appl., 29 ( 1997), 693715.

[3] E. C. Pielou, Mathematical Ecology, John Wiley, New York, 1977. [4] V. Volterra, Variazioni e fluttuazioni del numero d' individui in specie animali conviventi, Mem. R. Accad. Naz. Lincei, 2-3 (1926), 30-111. [5] A. J. Lotka, Elements of Physical Biology, William and Wilkins, Baltimore, 1925.

5 Limit Cycles

Aims and Objectives • To give abrief historical background. • To define features of phase plane portraits. • To introduce the theory of planar Iimit cycles. On completion of this chapter, the reader should be able to • prove existence and uniqueness of a Iimit cycle; • prove that certain systems have no Iimit cycles; • interpret Iimit cycle behavior in physical terms. Limit cycles, or isolated periodic solutions, are the most common form of solution observed when modeling physical systems in the plane. Early investigations were concerned with mechanical and electronic systems, but periodic behavior is evident in all branches of science. The chapter begins with a historical introduction, and then the theory of planar Iimit cycles is introduced.

5.1

Historical Background

Definition 1. A Iimit cycle is an isolated periodic solution.

S. Limit Cycles

78

Limit cycles in planar differential systems commonly occur when modeling both the technological and natural sciences. Most of the early history in the theory of Iimit cycles in the plane was stimulated by practical problems. For example, the differential equation derived by Rayleigh in 1877 [3], related to the oscillation of a violin string, is given by

x + t: ( ｾ＠

(.ii -

1) x+ x = 0,

where x = ｾ＠ and x = ｾＮ＠ Let x = y. Then this differential equation can be written as a system of first-order autonomous differential equations in the plane (5.1)

x

= y, y = -x- f Ｈｾ

Ｒ

t) Ｍ

y.

A phase portrait is shown in Figure 5.1.

Figure 5.1: Periodic behavior in the Rayleigh system (5.1) when f = 1.0. Following the invention of the triode vacuum tube, which was able to produce stable self-excited oscillations of constant amplitude, van der Pol [4] obtained the following differential equation to describe this phenomenon X + t:(x 2 - l)i

+X

= 0,

which can be written as a planar system of the form

(5.2)

x

= y, y = -x- t:(x 2 -

l)y.

5.1. Historical Background

79

Figure 5.2: Periodic behavior for system (5.2) when f = 5.0. A phase portrait is shown in Figure 5.2. Perhaps the most famous class of differential equations that generalize (5.2) are those first investigated by Lienard in 1928 [5],

x+ f(x)i + g(x) = 0, or in the phase plane (5.3)

i = y,

y=

-g(x) - f(x)y.

This system can be used to model mechanical systems, where f (x) is known as the damping term and g(x) is called the restoring force or stiffness. Equation (5.3) is also used to model resistor-inductor-capacitorcircuits (see Chapter 1) with nonlinear circuit elements. Limit cycles of Lienard systems will be discussed in some detail in Chapters 10 and 12. Possible physical interpretations for Iimit cycle behavior of certain dynamical systems are listed below: • For predator-prey models, the populations of the two species oscillate in phase with one another and the system is robust (see Example 3 in Chapter4). • For mechanical systems, examples include the motion of simple nonlinear pendula, wing rock oscillations in aircraft ftight dynamics, and surge oscillations in axial ftow compressors, for example. • For periodic chemical reactions, examples include the Landoll clock reaction and the Belousov-Zhabotinski reaction (see Chapter 8).

S. Limit Cycles

80

• For electrical or electronic circuits, it is possible to construct simple electronic oscillators using a nonlinear circuit element; a Iimit cycle can be observed if the circuit is connected to an oscilloscope. Limit cycles are common solutions for all types of dynamical systems. Sometimes it becomes necessary to prove the existence and uniqueness of a Iimit cycle, as described in the next section.

5.2

Existence and Uniqueness of Limit Cycles in the Plane

To understand the existence and uniqueness theorem, it is necessary to define some features of phaseplane portraits. Assurne that the existence and uniqueness theorem from Chapter 1 holds for all solutions considered here. Definition 2. Ajlow on 9l 2 is a mapping rr : m2 ｾ＠

!R 2 suchthat

1. rr is continuous; 2. rr(x, 0)) = x for all x e !R 2 ;

3. rr(rr(x, tt), t2) = rr(x, t1

+ t2).

Definition 3. The trajectory (or orbit) through x is defined as y(x) = {rr(x, t) : t E lx}. where lx is the maximal interval of existence. The positive semiorbit is defined as y+(x) = {rr(x, t) : t > 0}. The negative semiorbit is defined as y-(x) = {rr(x, t) : t < 0}. Definition 4. The positive Limit set of a point x is defined as A +(x)

= {y : there exists a sequence tn ｾ＠

oo suchthat rr(x, t) ｾ＠

y}.

The negative Limit set of a point x is defined as A -(x) = {y: there exists a sequence tn ｾ＠

-oo suchthat rr(x, t) ｾ＠

y}.

In the phase plane, trajectories tend to a critical point, a closed orbit, or infinity. Definition S. A set S is invariant with respect to a flow if x E S implies that y(x) C S. A set S is positively invariant with respect to a flow if x E S implies that y+(x) C S. A set S is negatively invariant with respect to a flow if x E S implies that y-(x) c S. A general trajectory can be labeled y for simplicity. Definition 6. A Iimit cycle, say

r, is

5.2. Existence and Uniqueness of Limit Cycles in the Plane

81

• a stable Limit cycle if A + (x) = r for all x in some neighborhood; this implies that nearby trajectories are attracted to the Iimit cycle; • an unstable Iimit cycle if A- (x) = r for all x in some neighborhood; this implies that nearby trajectories are repelled from the Iimit cycle; • a semistable Limit cycle if it is attracting on one side and repelling on the other. The stability of Iimit cycles can also be deduced analytically using the Poincare map (see Chapter 9). The following example will be used to illustrate each of Definitions 1-6 above and 7 below.

Definition 7. The period T of a Iimit cycle is given by x(t)

= x(t + T).

Example 1. Describe some of the features for the following set of polar differential equations in terms of Definitions l-7:

(5.4)

r=

r(I- r)(2- r)(3- r),

iJ

= -1.

Solution. A phase portrait is shown in Figure 5.3. There is a unique critical point at

the origin since iJ is nonzero. There are three limit cycles that may be determined from the equation r 0. They are the circles of radii one, two, and three, all centered at the origin. Let r; denote the Iimit cycle of radius r i.

=

=

Figure 5.3: Three limit cycles for system (5.4). There is one critical point at the origin. If a trajectory starts at this point, it remains there forever (part 2 of Definition 2). A trajectory starting at (1, 0) will reach the point (-I, 0) when t1 = rr and the motion is clockwise. Continuing on this path foranother time interval t2 rr, the orbit retums to (I, 0). Using part 3 of Definition 2, one can write 1r(rr((l, 0), t1), t2) = rr((l, 0), 2rr) since the Iimit cycle is of period 2rr (see below). On the Iimit cycle r 1, both the positive and negative semiorbits lie on r 1•

=

82

5. Limit Cycles

= (i,

=

Suppose that P 0) and Q (4, 0) are two points in the plane. The Iimit setsaregiven by A +{P) = fJ, A -{P) = (0, 0), A +(Q) = f3,and A -{Q) = oo. The annulus A1 = {r E 9l 2 : 0 < r < 1} is positively invariant, and the annulus Az = {r E 9l 2 : 1 < r < 2} is negatively invariant. If 0 < r < 1, then > 0 and the critical point at the origin is unstable. If 1 < r < 2, then < 0 and r1 is a stable Iimit cycle. If2 < r < 3, then; > 0 and r 2 is an unstable Iimit cycle. Finally, if r > 3, then < 0 and f3 is a stable Iimit cycle. Iotegrate both sides of iJ = -I w.r.t. time to show that the period of all of the Iimit cycles is 2Tf.

r

r

r

The Poincare-Bendixson Theorem. Suppose that y+ is contained in a bounded region in which there arefinitely many critical points. Then A +(y) is either

• a single critical point; • a single closed orbit; • a graphic-critical points joined by heteroclinic orbits (see Chapter 6). Corollary. Let D be a bounded closed set containing no critical points and suppose that D is positively invariant. Then there exists a Iimit cycle contained in D. A proof of this theorem involves topological arguments and can be found in [1], for example. Example 2. By considering the flow across the reetangle with corners at {-I, 2), (1, 2), (I, -2), and ( -1, -2), prove that the following system has at leastone Iimit cycle

x= y -

(5.5)

8x3,

y = 2y -

4x - 2y 3 .

x

Solution. The critical points are found by solving the equations = y = 0. Set y = 8x 3 . Then y = 0 if x(l - 4x 2 + 256x 8) = 0. The graph of the function y = I - 4x 2 + 256x 8 is given in Figure 5.4(a). The graph has no roots and hence the origin is the only critical point. Linearize at the origin in the usual way. lt is not difficult to show that the origin is an unstable focus. Consider the flow on the sides of the given rectangle:

= 2, lx I ｾ＠ 1, y = -4x - 12 < 0. On y = -2, lxl ｾｉＬ＠ y = -4x + 12 > 0.

• On y •

x = y- 8 < 0. On x = -1, IYI:::: 2, y = y + 8 > 0.

• On x = 1, •

IYI

ｾ＠

2,

5.2. Existence and Uniqueness of Limit Cycles in the Plane

83

The ftow is depicted in Figure 5.4(b). The reetangle is positively invariant and there are no critical points other than the origin, which is unstable. Consider a small deleted neighborhood, say NE, around this critical point. For exarnple, the boundary of NE could be a small ellipse. On this ellipse, all trajectories will cross outwards. Therefore, there exists a stable Iimit eyele lying inside the reetangular region and outside of NE by the eorollary to the Poincare-Bendixson Theorem.

s J//_.',/"..-....-

ｊｬｾＮｲＭ

__

4

3

y

0

o o • ｯＧＭＮ｜ＬｾＱＫｬ＠

..

t

.........

•

t

..............

..

t

. . . . . .......

t

. . . . . ._ .

..............ＮＭｾ＠

....-.---

........

_......

......

...-......... ..-. . . . . . .

........ .......

...--

......

..._.....__......__

ＱＴＭｾＺＮＫ＠

............._,./lf .........._,./'1/l

-1 ｾＮＸ＠

.r 0.2 0.4 0.6 0.8 ｾＮＴ＠

I

4

.....ｾ＠

--.............. _._._........._........ __._.._............./' __._._......_,. /

.,' r".--....-.-.-

,r.,......-.-........-

--.............. -3 • '-"2 .......

2

lllt/"..-....-.-

........../'/ 1 n

(a)

ｾＬＮ｟＠ ＬＮ｟ｾＭ

3

" , ＬＮ｟ｾ＠

(b)

Figure 5.4: ( a) Polynomial of degree eight. (b) Flow aeross the reetangle for system (5.5). Definition 8. A planar simple closed eurve is ealled a Jordan curve.

Consider the system (5.6)

x=

P(x, y),

y=

Q(x, y),

where P and Q have continuous first-order partial derivatives. Let the vector field be denoted by X and let l/1 be a weighting faetor that is continuously differentiable. Recall Green 's Theorem, whieh will be required to prove the following two theorems. Green's Theorem. Let J be a Jordan curve offinite length. Suppose that P and

Q are two continuously differentiahte functions defined on the interior of J, say D. Then

5. Limit Cycles

84 {{ [ap ax

llo

+ aQ] ay

dxdy =

1.. Pdy- Qdx.

J'.,

Dulac's Criteria. Consideran annular region, say A, contained in an open set E. lf V.(l/JX) = div(l/JX) = !._(l/1 P)

ax

+ !_(l/IQ) ay

does not change sign in A, then there is at most one Iimit cycle entirely contained in A. Proof. Suppose that r, and f'z are Iimit cycles encircling K, as depicted in Figure 5.5, of periods T1 and Tz, respectively. Apply Green's Theorem to the region R shown in Figure 5.5.

f{ jjR

[ß(l/IP) dX

+ d(l/IQ)]dxdy= J..

Now on f'1 and f'z, i [[ [ß(l/1 P) ax

}}R

lf!Pdy-lf!Qdx+ { lf!Pdy-lf!Qdx

jL fr 2 - J. l/1 Pdy -l/1 Qdx - { l/1 Pdy -l/1 Qdx. }L fr1

dy

= P and y = Q, so

+ a(l/IQ)] dx dy ay

= foT2(l/JPQ -l/IQP)dt- foTt (l/IPQ -l/IQP)dt = 0, which contradicts the hypothesis that div(l/JX) f= 0 in A. Therefore, there is at most one Iimit cycle entirely contained in the annulus A.

Figure 5.5: Two Iimit cycles encircling the region K.

Example 3. Use Dulac's criteria to prove that the system (5.7)

i = -y

+ x(l- 2x 2 -

has a unique Iimit cycle.

3y 2 ),

y = x + y(l- 2x 2 -

3y 2 )

5.3. Nonexistence of Limit Cycles in the Plane

85

Solution. Convert to polar coordinates using the transformations

rr =XX+ yy,

r 2Ö =

xy- y.X.

Therefore, system (5.7) becomes

r = r(l -

2r 2

-

r 2 sin 2 8),

Ö = 1.

!,

Since iJ = 1, the origin is the only critical point. On the circle r = r= !sin 28). Hence r > 0 on this circle. On the circle r 1, r -1- sin 2 e. Hence f < 0 on this circle. If r ｾ＠ 1, then < 0, and if 0 < r ｾ＠ then > 0. Therefore, there exists a Iimit cycle in the annulus A = {r : < r < 1} by the corollary to the Poincare-Bendixson Theorem. Considertheannulus A.Nowdiv(X) =2(1-4r 2 -2r 2 sin 2 8).If! < r < 1, then div(X) < 0. Since the divergence of the vector field does not change sign in the annulus A, there is at most one Iimit cycle in A by Dulac's criteria. A phase portrait is given in Figure 5.6.

!. The index, say /x(C), is defined as ｾ･＠

Ix(C) = - ,

2rr

where ｾ･＠

is the overall change in the angle

e.

The above definition can be applied to isolated critical points. For example, the index of a node, focus, or center is +1 and the index of a col is -1. Theorem 1. The sum of the indices of the critical points contained entirely within a Iimit cycle is + I . The next theorem then follows. Theorem 2. A Limit cycle contains at least one critical point. When proving that a system has no Iimit cycles, the following items should be considered:

1. Bendixson's criteria; 2. indices; 3. invariant lines; 4. critical points. Example 4. Prove that none of the following systems have any Iimit cycles: (a)

x = 1 + y2 -

exy,

y=

xy + cos 2 y.

(b) x=y 2 -x,y=y+x 2 +yxl; (c) x = y

+ x 3, y = x + y + y3 ;

87

5.3. None:xistence of Limit Cycles in the Plane (d) x=2xy-2y4 ,y=x 2 -y 2 -xy3;

(e) x

= x(2- y- x), y = y(4x- x 2 -

3) given

t/1

= }y·

Solutions. (a) The system has no critical points and hence no Iimit cycles by Theorem 2. (b) The origin is the only critical point and it is a saddle point (or col). Since the index of a col is -1, there are no Iimit cycles from Theorem 1. (c) Find the divergence, div X = ｾ＠ + ｾ＠ = 3x 2 + 3y 2 + I ｾ＠ are no Iimit cycles by Bendixson's criteria.

0. Hence there

(d) Find the divergence, div X= ｾ＠ + ｾ＠ = -3x 2y. Now div X= 0 if either x = 0 or y = 0. However, on the line x = 0, x = -2y4 ::;: 0, and on the line y = 0, y = x 2 2!: 0. Therefore, a Iimit cycle must lie wholly in one of the four quadrants. This is not possible since div X is nonzero here. Hence there are no Iimit cycles by Bendixson's criteria.

x

y

(e) The axes areinvariant since = 0 if x = 0 and y = 0 if = 0. The weighted divergence is given by div(t/fX) = fx(t/IP) + ;y(t/IQ) = ＭｾＮ＠ Therefore, there are no Iimit cycles contained entirely in any of the quadrants, and since the axes are invariant, there are no Iimit cycles in the whole plane. E:xample 5. Prove that the system

x

= x(l -

4x + y),

y

= y(2 + 3x -

= xm yn. The axes are invariant since x = 0 on x = 0 and y = 0 on y = 0. Now

has no Iimit cycles by applying Bendixson 's criteria with Solution.

2y)

div(t/fX)

= aax

t/1

(xm+iyn -4xm+2yn +xm+iyn+i)

+ :y (zxmyn+i + 3xm+lyn+l _ 2xmyn+2), which simplifies to div(t/IX) = (m+2n+2)xmyn +(-4m+3n -5)xm+i yn +(m -2n -3)xmyn+l. Select m

= ｾ＠

and n

= -l Then . dtv{t/fX)

43 3 =-4 x'2 ｹＭｾＮ＠

5

Therefore, there are no Iimit cycles contained entirely in any of the four quadrants, and since the axes are invariant, there are no limit cycles at all.

88

5. Limit Cycles

5.4

Maple Commands

The Maple commands below may be edited to produce solutions and diagrams for all of the examples and exercises appearing in Chapter 5. >with(DEtools); >sys1:=diff(x(t),t)=y(t), diff(y(t),t)=-x(t)-5.0*y(t)*((x(t))A2-1); >DEplot ( [sys1], [x(t) ,y(t) I, t=O .. 30, [ [0, 2 ,1] ), stepsize=O. 01,x=-3 .. 3, >y=-10 .. 10); ># See Figure 5.2. > >sys2:=diff(x(t),t)=y(t)+x(t)*(1-sqrt((x(t))A2+(y(t)) A2)) >*(2-sqrt((x(t))A2+(y(t))A2)), diff(y(t),t)=-x(t)+y(t) >*(1-sqrt((x(t))A2+(y(t))A2))*(2-sqrt((x(t))A2+(y(t) )A2)); >DEp1ot ( [sys2], [x(t) ,y(t) ], t=O .. 10, [ [0,1, 0], [0, 2,0]], >stepsize=0.01,x=-3 .. 3,y=-3 .. 3); ># See Figure 5.3. >

>with(p1ots): >plot(256*xA8-4*xA2+l,x=-1 .. l,y=0 .. 4); >#See Figure 5.4(a). > >dfieldplot([diff(x(t),t)=y(t)-8*(x(t))A3, >diff(y(t),t)=2*y(t)-4*x(t)-2*(y(t))A3], >[x(t),y(t)),t=-2 .. 2,x=-2 .. 2,y=-3 .. 3,arrows=SLIM); ># See Figure 5.4(bl. > >solve({yA2-x,y+xA2+y*xA3},{x,y}); X=

y= 0

0,

>P:=y+XA3: >Q:=x+y+yA3: >divX:=diff(P,x)+diff(Q,y);

div X = 3x 2 + 3i + 1

5.5 Exercises I. Prove that the system

x = y+ x ( ｾ＠

- x 2 - y 2)

,

y=

-x + y ( l - x2 -

i)

has a stable Iimit cycle. 2. By considering the flow across the square with coordinates (1, 1), (1, -1), (-1, -1), ( -1, 1), centered at the origin, prove that the system

x = -y + xcos(rrx), has a stable Iimit cycle.

y = x- y 3

89

S.S. Exercises 3. Prove that the system

X=X

-

y - x 3,

y=X + y -

y3

has a unique Iimit cycle. 4. Prove that the Holling-Tanner model

x.

= xß ( I - -x) k

rxy , (a+ax)

.

y=by

(I--;Ny)

has at most one Iimit cycle.

5. Prove that none of the following systems have Iimit cycles: (a) .i

= y, y = -x -

(I

+ x 2 + x 4 )y;

(b) i=x-x 2 +2y 2 ,y=y(x+l);

.i = y 2 - 2x, y = 3- 4y- 2x 2 y; (d) .i = x(4 + 5x + 2y), y = y(-2 + 1x (c)

,,, -

'I'-

I .

+ 3y), using the multiplier

XJ!•

(e) .i = x(ß -8x- yy), y

= y(b-dy -cx), using the multiplier 1/1 = }y.

In case (e), prove that there are no Iimit cycles in the first quadrant only; these differential equations were used as a general model for competing species in Chapter 4.

Recommended Reading [1] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, Berlin, New York, Heidelberg, 1993. [2] Ye Yan Qian, Theory of Limit Cycles, Translations of Mathematical Monographs 66, American Mathematical Society, Providence, 1986. [3] J. Rayleigh, The Theory ofSound, Dover, New York, I945. [4] A. Lienard, Etude des oscillations entrenues, Rev. Generale de l'Electricite, 23 (1928), 946-954. [5] B. van der Pol, On relaxation oscillations, Philos. Magazine, 7 (1926), 901912, 946-954.

6 Hamiltonian Systems, Lyapunov Functions, and Stability

Aims and Objectives • To study Hamiltonian systems in the plane. • To investigate stability using Lyapunov functions. On completion of this chapter, the reader should be able to • prove whether or not a system is Hamiltonian; • sketch phase portraits of Hamiltonian systems; • use Lyapunov functions to determine the stability of a critical point; • distinguish between stability and asymptotic stability. The theory of Hamiltonian (or conservative) systems in the plane is introduced. The differential equations are used to model dynamical systems in which there is no energy loss. Hamiltonian systems arealso used extensively when bifurcating Iimit cycles in the plane (see Chapters 12 and 13). Sometimes it is not possible to apply linearization techniques to determine the stablility of a critical point. In certain cases, the ftow across Ievel curves, defined by Lyapunov functions, can be used to determine the stability.

92

6.1

6. Hamiltonian Systems, Lyapunov Functions, and Stability

Hamiltonian Systems in the Plane

Definition 1. A system of differential equations on m2 is said to be Hamiltonian with one degree of freedom if it can be expressed in the form (6.1)

aH dx -=-, ay dt

an

dy

-=--, ax dt

where H (x, y) is a twice-continuously differentiable function. The system is said to be conservative and there is no dissipation. In applications, the Hamiltonian is defined by H(x, y) = K(x, y) + V(x, y), where K is the kinetic energy and V is the potential energy. Hamiltonian systems with two degrees of freedom will be discussed in Chapter 9. Theorem 1 (Conservation of Energy). The total energy H (x, y) isafirstintegral and a constant of the motion. Proof. The total derivative along a trajectory is given by

dH dt

= aH dx + aH dy = 0 ax dt

ay dt

from the chain rule and (6.1 ). Therefore, H (x, y) is constant along the solution curves of (6.1), and the trajectories lie on the contours defined by H(x, y) = C, where C is a constant. Consider a simple mechanical system which is Hamiltonian in the plane. The Simple Nonlinear Pendulum. The differential equation used to model the motion of a pendulum in the plane may be derived using Newton's law ofmotion: (6.2)

d 2e

g

=0 -dt2 + -sine ' I

where e is the angular displacement from the vertical, 1 is the length of the arm of the pendulum, which swings in the plane, and g is the acceleration due to gravity. This model does not take into account any resistive forces, so once the pendulum is set into motion, it will swing periodically forever, thus obeying the conservation of energy. The system is called conservative since no energy is lost. A periodically forced pendulum will be discussed in Chapter 9. Let iJ = t/J. Then system (6.2) can be written as a planar system in the form (6.3)

. t/>

g

= --1 sine.

The critical points occur at (mr, 0) in the (0, t/>) plane, where n is an integer. It is not difficult to show that the critical points are hyperbolic if n is odd and nonhyperhoHe if n is even. Therefore, Hartman 's Theorem cannot be applied when

93

6.1. Hamiltonian Systems in the Plane

Pivot

9

Figure 6.1: A simple nonlinear pendulum.

n is even. However, system (6.3) is a Hamiltonian system with H(O, f/J) =

ｾ＠

-

f cos e (kinetic energy + potential energy), and therefore the solution curves may

be plotted. The direction field may be constructed by considering ｾＮ＠ 8, and tjJ. Solution curves and direction fields are given in Figure 6.2(a). The axes of Figure 6.2(a) are the angular displacement (0) and angular velocity (0). The closed curves surrounding the critical points (2mr, 0) represent periodic oscillations, and the wavy lines for large angular velocities correspond to motions in which the pendulum spins around its pivotal point. The closed curves correspond to local minima on the surface z = H (0, f/J ), and the unstable critical points correspond to local maxima on the same surface. Use the left button in Maple to rotate the surface.

Definition 2. A critical point of the system (6.4)

at which the Jacobian matrix has no zero eigenvalues is called a nondegenerate critical point; otherwise, it is called adegenerate critical point.

Theorem 2. Any nondegenerate critical point of an analytic Hamiltonian system is either a topological saddle point or a center.

Proof. Assurne that the critical point is at the origin. The Jacobian matrix is equal to

94

6. Hamiltonian Systems, Lyapunov Functions, and Stability

(b)

(a)

Figure 6.2: (a) Phase portrait for system (6.3) when -4;rr surface z = H((}, 0 such that lim llx(t)- xo(t)ll = 0 ,_00 whenever llx(to)- xo(to)ll < 11· A trajectory near a stable critical pointwill remain close tothat point, whereas a trajectory near an asymptotically stable critical pointwill move closer and closer to the critical point as t --+- oo.

98

6. Hamiltonian Systems, Lyapunov Functions, and Stability

-...'-.'.\\

J

I ."o/r-

11>""".., /I Ｏｲ Ｍ Ｎ ｾ＠ ｩｬｯＯｲＭｾ＠ /I",.-..-..-

"""'''-\\ -.-....'-.\\ ·-.--.'-,\ \ . . . . . . . '* '\ \ ........, \

/ .".--..-·.-·

-·-·--'- \

I-"""·-.Ir.-..........

/ ................... .........

(a)

(b)

Figure 6.5: (a} Phaseportrait for Example 2. (b) The double-weil potential. The following theorem holds for system (6.4) when X are given in Chapter 8.

e mn. Examples in m3

The Lyapunov Stability Theorem. Let E be an open subset of !R" containing an isolated critical point xo . Suppose that f is continuously dijjerentiable and that there exists a continuously differentiablefunction, say V(x), that sarisfies the following conditions: • V(xo) = 0; • V(x) > 0

where

X E

ifx ::P xo.

m". Then

I.

if V(x) :S 0 for alt x

2.

if V(x)

E E, xo is stable;

< 0 for all x E E, xo is asymptotically stable;

3. ifV(x) > Oforallx

E

E, xo is unstable.

Proof. I. Choose a small neighborhood N* surrounding the critical point XQ. In this neighborhood, V(x) :::: 0, so a positive semiorbit starting inside N* remains there forever. The same conclusion is drawn no matter how small E is chosen to be. The critical point is therefore stable.

99

6.2. Lyapunov Functions and Stability

2. Since V(x) < 0, the Lyapunov function must decrease monotonically on every positive semiorbit x(t). Let t/>r define the flow on (6.4). Then either V(t/>r) ｾ＠ xo as t ｾ＠ oo or there isapositive semiorbit x(t) suchthat (6.5)

V ( t/>r) :::::: n > 0

for all t :::::: to

for some n > 0. Since xo is stable, there is an annular region A, defined by n =::: V(x) =::: c, containing this semiorbit. Suppose that V attains its upper bound in A, say -N, so V(x) =::: -N < 0,

x e A,

N > 0.

Integration gives V(x(t))- V(x(to)) =::: -N(t- to), where t > to. This contradicts (6.5), and therefore no path fails to approach the critical point at xo. The critical point is asymptotically stable. 3. Since V(x) > 0, V(x) is strictly increasing along trajectories of (6.4). lf tf>t is the flow of (6.4), then V(t/>r) > V(xo) > 0 fort> 0 in a small neighborhood ofxo. NE. Therefore, V(t/>r)- V(xo):::::: kt

for some constant k and t :::::: 0. Hence for sufficiently large t, V(t/>1 ) > kt > K,

where K is the maximum ofthe continuous function V(x) on the compact set NE. Therefore, t/>r lies outside the closed set NE and xo is unstable. Definition 8. The function V(x) is called a Lyapunov function. Unfortunately, there is no systematic way to construct a Lyapunov function. The Lyapunov functions required for specific examples will be given in this book. Note that if V(x) = 0, then all trajectories lie on the curves (surfaces in mn) defined by V (x) = C, where C is a constant. The quantity V gives the rate of change of V along trajectories, or in other words, V gives the direction that trajectories cross the Ievel curves V(x) = C. Example 3. Determine the stability of the origin for the system

x=-i.

•

3

y=x.

6. Hamiltonian Systems, Lyapunov Functions, and Stability

100

Solution. The eigenvalues are both zero and the origin is a degenerate critical point. A Lyapunov function for this system is given by V (x, y) = x4 + y4 , and

furthermore, av dx av dy 3 3 3 3 -dV = +-= 4x (-y ) + 4y (x ) = 0. dt ax dt ay dt

Hence the solution curves lie on the closed curves given by x 4 + y4 = C. The origin is thus stable but not asymptotically stable. The trajectories that start near to the origin remain there but do not approach the origin asymptotically. If y > 0, then x < 0, and if y < 0, then x > 0. The Ievel curves and direction fields are given in Figure 6.6.

Figure 6.6: Phase portrait for Example 3. Example 4. Investigate the stability of the origin for the system

x = y,

y = -x -

y(l - x 2)

using the Lyapunov function V(x, y) = x 2 + y2. Solution. Now -dV dt

so

avdx avdy - + - - = 2x(y)+2y(-x- y+ yx 2), =ax dt ay dt dV 2 2 - = 2y (x -I) dt

101

6.3. Maple Commands

and V :::: 0 if lxl :::: 1. Therefore, V= 0 if either y = 0 or x = ±1. When y = 0, = 0 and y = -x, which means that a trajectory will move off the line y = 0 when x ::/= 0. Hence if a trajectory starts inside the circle of radius 1 centered at the origin, then it will approach the origin asymptotically. The origin is asymptotically stable.

x

Example 5. Prove that the origin of the system

. = - 8X X

3 -xy 2 - 3y,

is asymptotically stable using the Lyapunov function V(x, y) = 2x 2 + 3y 2. Oetermine the Lyapunov domain of stability. Solution. Now

V=

4x( -Sx - xi - 3y 3)

+ 6y(2x 2y + 2xi) =

8x 2 (y 2 - 4)

0 if IYI :::: 2. Therefore, V = 0 if either x = 0 or y = ±2. When -3y 3 and y 0, which means that a trajectory will move offthe line x = 0, x = 0 when y ::/= 0. Now V < 0 if IYI < 2. This implies that V < 0 as long as V(x, y) = 2x 2 + 3y 2 < 12. This region defines the domain ofLyapunov stability. Therefore, if a trajectory lies wholly inside the ellipse 2x 2 + 3y 2 = 12, it will move to the origin asymptotically. Hence the origin is asymptotically stable. A phase plane diagram and the Lyapunov domain of stability is given in Figure 6.7. An approximation of the true domain of stability for the origin of the system in Example 5 is indicated in Figure 6.7(a). Notice that it is !arger than the Lyapunov domain of stability and that the x axis is invariant.

V ::::

and

x=

6.3

=

Maple Commands

The Maple commands below may be edited to produce solutions and diagrams for all of the examp1es and exercises appearing in Chapter 6. >wi th ( linalg) : >with(DEtools): >with(plots): >

>solve({y+xA2-yA2,-x-2*x* y},{x,y})

{x = 0, y

= 0},

>J: =rna trix ( [ [ 0,

{x

= 0, y =

1}, {x

= ｾｲｯｴｦＨＭＳ＠

+ Z 2 ), y = Ｍｾｽ＠

-1 J , [- 3, 0] ] ) ;

>eigenvects (J);

[- -13, 1, {[1, -13]}]. [-13. 1, {[1, -v'3]}]

102

6. Hamiltonian Systems, Lyapunov Functions, and Stability

(b)

(a)

Figure 6.7: (a) Phaseportrait for Example 5. (b) The domain ofLyapunov stability. >sys:=diff(x(t),t)=y(t),di ff(y(t),t)=x(t)+(x(t))A2; >DEplot ( [sys], [x(t) ,y(t) l, t:-10 .. 10, > ( ( 0, 1, 0 l, ( 0, 0.1, 0 .1] , ( 0' 0 .1' -0. 1] , [ 0, 1. 2, 01 ' [ 0, -0.1, 0. 1] , [ 0, 0, 1]] , >stepsize=0.05,x=-2 .. 2,y=-2 .. 2); ># See Figure 6.3. >implicitplot(2*xA2+3*yA2= 12,x=-4 .. 4,y=-4 .. 4); >#See Figure 6.7(b).

6.4 Exercises 1. Find the Hamiltonian of the system

x = y,

y = x- x 3

and sketch a phase portrait. 2. Given the Hamiltonian function H (x, y) portrait for the Hamiltonian system.

=f +f

-x

3. Plot a phase portrait for the damped pendulum equation

Ö + 0.150

+ sin 8 = 0

and describe what happens physically.

4 4 ,

sketch a phase

103

6.4. Exercises

4. Investigate the stability of the critical points at the origin for the following systems: (a) i = - y - x3, x2 + y2;

y=x

(b) i = x(x- a), 0, there are two critical points, one at 0 = (0, 0) and the other at B = (Jl.., 0). The origin is now a saddle point and B is a stable node. A vector field and manifolds are plotted in Figure 7.3(c).

l\\ ｉ｜ＱＯｕｎｾＭ ..-;' ,/ ll \\\ ..-;'./II \ \ \

;'.t'/li ;';'//

..-..- 0, separately. Case (i). When J.t < 0, there is one critical point at 0 = (0, 0). The origin is a stable node. A vector field and the manifolds at the origin are shown in Figure 7.5(a). Case (ii). When J1.. = 0, there is one nonhyperbolic critical point at the origin. The solution curves satisfy the differential equation

110

7. Bifurcation Theory

[x

ol---J /

/

/

/

/

/

/

/

0 Figure 7.4: A bifurcation diagram for system (7.3) showing a transcritical bifurcation. The solid lines depict stable behavior and the dashed lines depict unstable behavior. Ｍｾ｜ａｊｉＯｙ

ﾷｾＭ｜ｊｉＯＮ

.•.

ﾷｾ｜ＱＮＭ

•••

• • • ﾷｾ｜ｊｉ＠

........ ......-.-. ••

ｾ｜Ｏｙﾷ＠

II Ｏｾ＠ ｾ＠ \\\ ｾＭ ｉｙｾＭ --.......-.-...\ 11/J\\\\ ［ＮＭｾ ----.-...'. I/I J \\'.\ Ｏｙｾ＠ Ｍｾ｜＠

•••

\ ',!' . . . . -. . . . . . .

....... '.......'-. \

/.-"y.-

,....... \ ;....-..-

....

Ｍｾ｜＠

-----.-...'. //II \\'.'. /..-.-.-.｜ＧＮｾ＠ ｾＯ＠ Ｍｾ＠ .,-ｾＭ Ｍｾ＠

..-

,,

._

X

__.,..I ........... __.__.",., ............

....... , ........_._"."..,.,' \\, ••••• .._........ I' I

........... _._....,.../, \, \ ........_............

...... _.__.,../I'\,, ......_... '''

11/J\\\\

_.._._._.",. _.._._....../

____.__./'

ｾＮＭＬ［＠

｜ｾ＠

＾ＢＧｾＯ＠

____

｜ｾＭＮ＠

............... ,1 . . . . . . . . . ,,,, ,,,\

_ , ...._.._._._.__

/?",.",. /I'?/' ,..__.._.__

,..._.._.__ ,._

___._......__.",.1'/1\\'-'._...._,__,__

''-\\/11'1' ,,,\!11'1 _ _.,..I \\\\!111 ......_.._..._ __......,..I \\\\f/11 ,...._...._,__ __.__..,..../ \\\if/11 ,...._._._.__

(a)

(b)

_ _

........_

_.__._..../lf\\'-..0....._..______

_._.._.__./'

Figure 7.5: Vector field plots and manifolds when (a) 11dy

dx which has solutions IYI in Figure 7.5(a).

I

ｾ＠

0 and (b) tL > 0.

y

= x3 '

= K e -b"Z, where K is a constant. A vector field is plotted

7.1. Bifurcations ofNonlinear Systems in the Plane

111

Case (iii). When J.L > 0, there are three critical points at 0 = (0, 0), A = (,jii, 0), and B ,jii, 0). The origin is now a saddle point and A and B are both stable nodes. A vector field and all of the stable and unstable manifolds is plotted in Figure 7.5(b). The behavior of the critical points can be summarized on a bifurcation diagram, as depicted in Figure 7.6.

= (-

X

0

0 Figure 7.6: A bifurcation diagram for system (7 .4) showing a pitchfork bifurcation. The solid curves depict stable behavior and the dashed curves depict unstable behavior. Note the resemblance of the stable branches to a pitchfork.

7.1.IV A Hopf Bifurcation. Consider the system (7.5) The origin is the only critical point since iJ :F 0. There are no Iimit cycles if (i) J.L ｾ＠ 0 and one if (ii) 1.1.. > 0. Consider the two cases separately. Case (i). When 1.1.. ｾ＠ 0, the origin is a stable focus. Since iJ < 0, the flow is clockwise. A phase portrait and vector field is shown in Figure 7 .7(a). Case (ii) When 1.1.. > 0, there is an unstable focus at the origin and a stable Iimit cycle at r = ...(iJ, since r > 0 if 0 < r < ...(iJ, and r < 0 if r > ...(ii.. A phase portrait is shown in Figure 7.7(b). The qualitative behavior can be summarized on a bifurcation diagram, as shown in Figure 7 .8. As the parameter J.L passes through the bifurcation value J.Lo = 0 a Iimit cycle bifurcates from the origin. The amplitude of the Iimit cycle grows as J.L increases. Think of the origin blowing a smoke ring.

7. Bifurcation Theory

112

(b)

(a)

Figure 7.7: Phaseportraits when (a) J1 ｾ＠

0 and (b) 11 > 0.

r

0 0 Figure 7.8: A bifurcation diagram for system (7 .5) showing a Hopfbifurcation. The solid curves depict stable behavior and the dashed curves depict unstable behavior.

7.2

Multistability and Bistability

There are two types of Hopf bifurcation, one in which stable Iimit cycles are created about an unstable critical point, called the supercritical Hopf bifurcation (see Figure 7.8), and the other in which an unstable Iimit cycle is created about a stable critical point, called the subcritical Hopfbifurcation (see Figure 7.9). In the engineering literature, supercritical bifurcations are sometimes called soft (or safe ); the amplitude of the Iimit cycles build up gradually as the parameter, 11 in this case, is moved away from the bifurcation point. In contrast, subcritical bifurcations are hard (or dangerous). A steady state, say at the origin, could become unstable as a parameter varies and the nonzero solutions could tend to infinity. An example ofthistype of behavior can be found in Figure 7.9. As J1 passes through zero from positive to negative values, the steady-state solution at the origin becomes

113

7.2. Multistability and Bistability

r

0

I

I

I'

/

----------0

Figure 7.9: A bifurcation diagram showing a subcritical Hopf bifurcation. The solid curves depict stable behavior and the dashed curves depict unstable behavior. unstable and trajectories starting anywhere other than the origin would tend to infinity. It is also possible for Iimit cycles of finite amplitude to suddenly appear as the parameter JL is varied. These Iimit cycles are known as large-amplitude Iimit cycles. Examples of this type of behavior include surge oscillations in axial flow compressors and wing rock oscillations in aircraft flight dynamics; see [1] for examples. Unstable Iimit cycles arenot observed in physical applications, so it is only stable large-amplitude Iimit cycles that are of interest. These Iimit cycles can appear in one of two ways: either there is a jump from a stable critical point to a stable large-amplitude Iimit cycle or there is a jump from one stable Iimit cycle to another of !arger amplitude. These bifurcations are illustrated in the following examples. Large-Amplitude Limit Cycle Bifurcations. Consider the system (7.6) The origin is the only critical point since (J ::f= 0. This critical point is stable if JL < 0 and unstable if JL > 0. The system undergoes a subcritical Hopfbifurcation at JL = 0 as in Figure 7.9. However, the new feature here is the stable largeamplitude Iimit cycle that exists for, say, JL > JLs. In the range JLs < r < 0, there exist two different steady-state solutions; hence system (7.6) is multistable in this range. The choice of initial conditions determines which steady state will be approached as t--+- oo. Definition 2. A dynamical system, say (7.1), is said tobe multistable if there is more than one possible steady-state solution for a fixed value of the parameter JL. The steady state obtained depends on the initial conditions. The existence of multistable solutions allows for the possibility of bistability (or hysteresis) as a parameter is varied. The two essential ingredients for bistable behavior are nonlinearity and feedback. To create a bistable region, there must

114

7. Bifurcation Theory

be some history in the system (see Chapter 16). Suppose that the parameter J.L is increased from some value less than J.Ls. The steady state remains at r = 0 until J.L = 0, where the origin loses stability. There is a suddenjump (a subcritical Hopf bifurcation) to the large-amplitude Iimit cycle, and the steady state remains on this cycle as J.L is increased further. If the parameter J.L is now decreased, then the steady state remains on the large-amplitude Iimit cycle until J.L = J.Ls, where the steady state suddenly jumps back to the origin (a saddle-node bifurcation) and remains there as J.L is decreased further. In this way, a bistable region is obtained, as depicted in Figure 7 .10.

r

Figure 7.10: A bifurcation diagram depicting bistable behavior for system (7 .6).

Definition 3. A dynamical system, say (7 .1 ), has a bistable solution if there are two steady states for a fixed parameter J.L and the steady state obtained depends on the history of the system. Now consider the system (7.7) A bistable region may be obtained by increasing and then decreasing the parameter J.L as in the example above. A possible bifurcation diagram is given in Figure 7 .11. In this case, there is a supercritical Hopf bifurcation at J.L = 0 and saddle-node bifurcations at J.LB and J.LA, respectively. Jumps between different steady states have been observed in the real world in wing rock phenomena and surge in jet engines, for example. Parameters need to be chosen that avoid such large-amplitude Iimit cycle bifurcations, and research is currently under way in this respect. Bistability also has many positive applications in the real world; for example, nonlinear bistable optical resonatorswill be investigated in Chapters 16 and 17. The author is also currently carrying out research with an astrophysicist to model bistability and instability in cloud nebulae.

115

7.3. Maple Commands

r

ｯｾＭ

(I

Figure 7.11: A bifurcation diagram depicting bistable behavior for system (7.7).

7.3 Maple Commands The Maple commands below may be edited to help with the understanding of Chapter 7. To save time and effort, it is better to use the dfieldplot command when plotting the phase portraits; this also overcomes the problems with DEplot. Animations are used to see how functions change as a parameter is varied. >with(DEtools): >wi th (plots) : >

>mu:=O: ># Try mu:=-2 and mu:=2 also. ＾､ｦｩ･ｬｰｯｴＨ｛ｸＩＬ］ｭｵＭｾＲｹ｝＠

>[x(t) ,y(t) ],t=-2 .. 2, >x=-4 .. 4,y=-4 .. 4,arrows=SLIM); >#See Figure 7.l(b). >

>?animate >mu:='mu•: ＾｡ｮｩｭｴ･Ｈ｛ｸＬｵＭｾＲ］Ｔ＠ .. 4],mu=-4 .. 4,numpoints=lOO,frames=100); ># Click on the graph and press the play button. ># This animation shows how many critical points there are for mu >from -4 to 4. ># See the saddle-node bifurcation diagram, Figure 7.2.

>

.. 2],mu=-0.5 .. 0.5,numpoints=l00, ＾｡ｮｩｭｴ･Ｈ｛ｲＬＪＭＱＩｾＲｵ］Ｐ＠

>frames=lOO); >#See Exercise 3(b).

7.4 Exercises 1. Consider the following one-parameter families of first-order differential equations defined on m:

ll6

7. Bifurcation Theory (b) ( )

C

x = X(J.L + ex); •

IJ.X

X= X - l+x2·

Detennine the critical points and the bifurcation values, plot vector fields on the line, and draw a bifurcation diagram in each case.

x

2. Use the animate command in Maple to show how varies as J.L increases from -4 to +4 for each of the differential equations in Exercise 1. 3. Construct first-order ordinary differential equations that have the following: (i) three critical points (one stable and two unstable) when J.L < 0, one critical point when J.L 0, and three critical points ( one unstable and two stable) when J.L > 0;

=

(ii) two critical points (one stable and one unstable) for J.L ::/:. 0 and one critical point when J.L = 0;

(iii) one critical point if IJ.LI :=:: 1 and three critical points if IJ.LI < 1. Draw a bifurcation diagram in each case. 4. A certain species of fish in a large Iake is harvested. The differential equation used to model the population, x(t) in hundreds of thousands, is given by dx

dt =

X

(

I-

x)

S -

hx

0.2 +X •

Detennine and classify the critical points and plot a bifurcation diagram. How can the model be interpreted in physical tenns? 5. Consider the following one-parameter systems of differential equations: (a)

x = x, y = J.L- y4;

(b) x = x 2 - XJ.L 2, y = -y; (c)

x = -x 4 + 5J.Lx 2 -

4J.L 2,

y = -y.

Find the critical points, plot phase portraits, and sketch a bifurcation diagram in each case. 6. Consider the following one-parameter systems of differential equations in polar form: (a)

(b)

r = J.Lr(r + J.L) 2 ,(; = I; r = r(J.L- r)(J.L- 2r), iJ = -1;

(c) ; = r(J.L2- ,z}, iJ = 1.

117

7.4. Exercises

Plot phase portraits for JL < 0, JL = 0, and JL > 0 in each case. Sketch the corresponding bifurcation diagrams. 7. Plot a bifurcation diagram for the planar system

; = r

(J.L -

0.2r 6 + r 4

-

r 2) ,

iJ

= -1

and indicate the regions where the system is multistable and/or possibly bistable. 8. Plot a bifurcation diagram for the system

; = r ( (r - 1) 2 - J.Lr) ,

iJ = 1.

Give a possible explanation as to why this type of bifurcation should be known as afold bifurcation. 9. Show that the one-parameter system

X = y + JLX

- X y2 '

undergoes a Hopf bifurcation at J.Lo bifurcation diagram.

•

3

Y = ILY -x- Y

= 0. Plot phase portraits and sketch a

10. Thus far, the analysis has been restricted to bifurcations involving only one parameter, and these are known as codimension-1 bifurcations. This example illustrates what can happen when two parameters are varied, allowing socalled codimension-2 bifurcations. The following two-parameter system of differential equations may be used to model a simple Iaser: i

= x(y- 1), y = a + ßy- xy.

Find and classify the critical points and sketch the phase portraits. Illustrate the different types of behavior in the (a, ß) plane and determine whether or not any bifurcations occur.

Recommended Reading [1] S. Lynch and C. J. Christopher, Limit cycles in highly nonlinear differential equations, J. Sound Vibration, 224-3 (1999), 505-517. [2] G Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory, Springer-Verlag, Berlin, New York, Heidelberg, 1997. [3] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, Springer-Verlag, Berlin, New York, Heidelberg, 1994. [4] S. H. Strogatz, Nonlinear Dynamics and Chaos withApplications to Physics, Biology, Chemistry, andEngineering ,Addison-Wesley, Reading, MA, 1994.

8 Three-Dimensional Autonomous Systems and Chaos

Aims and Objectives • To introduce first-order ordinary differential equations in three variables. • To introduce the notion of chaos. On completion of this chapter, the reader should be able to • construct phase portraits for linear systems in three dimensions; • use the Maple package to plot phase portraits and time series for nonlinear systems; • identify chaotic solutions; • interpret the solutions to modeling problems taken from meteorology, electric circuits, and chemical kinetics. Three-dimensional autonomous systems of differential equations are considered. Critical points and stability are discussed and the concept of chaos is introduced. Examples include the Lorenz equations, used as a simple meteorological model and in the theory of Iasers; Chua's circuit, used in nonlinear electronics and radiophysics; and the Belousov-Zhabotinski reaction, used in chemistry and biophysics. All of these systems can display highly complex behavior that can be interpreted from phase portrait analysis or Poincare maps.

8. Three-Dimensional Autonomous Systems and Chaos

120

Basic concepts are explained via example rather than mathematical rigor. Strange or chaotic attractors are constructed using the Maple package, and the reader is encouraged to investigate these systems through the exercises at the end of the chapter. Chaos will also be discussed in other chapters of the book.

8.1

Linear Systems and Canonical Forms

Consider linear three-dimensional autonomous systems of the form

x = a11x + a12y + a13z,

+ a22Y + a23Z, z= a31X + a32y + a33Z,

Y= (8.1)

a21X

where the aij are constants. The existence and uniqueness theorem (see Section 1.4) holds, which means that trajectories do not cross in three-dimensional space. The real canonical forms for 3 x 3 matrices are

,, =

h=

C' C' ｾ＠

ｾ＠

0

A2

ｾ＠

0

AJ

AI

ｾ＠ A2

0

) J,=O ) C' '

'

]4

=

ｾ＠

-ß Ct

0

AI 0

Matrix J 1 has three real eigenvalues, matrix h has a pair of complex eigenvalues, and matrices 1) and J4 have repeated eigenvalues. The type of phase portrait is determined from each of these canonical forms. Consider the following examples. Definition 1. Suppose that p e ffi 3 is a critical point of the system i = f(x), where x e 9! 3 . Then the stable and unstable manifolds of the critical point p are defined by Ws(p) = {x: A +(x) = p}, Wu(P) = {x: A -(x) = p},

Example 1. Solve the following system of differential equations, sketch a phase portrait, and define the manifolds:

X =X, (8.2)

y=y, z= -z.

Solution. There is one critical point at the origin. Each differential equation is integrable with solutions given by x(t) = c, e', y(t) = C2e', and z(t) = C3e-t.

8.1. Linear Systems and Canonical Forms

121

=

TheeigenvaluesandcorrespondingeigenvectorsareA1,2 1, (0, 1, O)T, (1, 0, O)T and AJ = -1, (0, 0, l)T. System (8.2) may be uncoupled in any of the xy, xz, or yz planes. Planar analysis gives an unstable singular node in the xy plane and cols in each ofthex z and yz planes. The phaseplane portraits for two of the uncoupled systems are given in Figure 8. t. If z > 0, i < 0, and if z < 0, i > 0. Putting all of this together, any trajectories not lying on the manifolds ftow along "lampshades" in three-dimensional space, as depicted in Figure 8.2.

\\\\\\\\\

\\\\\\\\\ \\\\\\\\'-

ｾＧＢ＠

\\\\\\\'-'\\\\\\'-'-"

,, \/I/

Ｍｾ＠

. . . . . . . . . ,,,, \

I //////;

,,,,,,,,, ,,,,,,,,, \\\\\\\\\ \\\\\\\\\ \\\\\\\\

l/!111/// l/1111/// ll/11111/ l/1111111 111111111

/1////// /11/////

............, , , , , , ,,,,,,,,

(a)

(b)

Figure 8.1: Phase plane portraits in the (a) xy and (b) xz planes. (a) is an unstable planar manifold. The z-axis is a one-dimensional stable manifold since trajectories on this line are attracted to the origin as t -+ +oo. The xy plane is a two-dimensional unstable manifold since all trajectories in this plane are attracted to the origin as t -+ -oo.

Example 2. Given the linear transformations x - Yl + z1, show that the system

Xi = -3Xl + lOyl' YI = -2xl +5yJ, ZI = -2XJ

+ 2yl + Jz1

can be transformed into

x =x -2y, y=

(8.3)

= x1 -

2x + y,

z= 3z.

2y 1, y

= -y1, and z =

122

8. Three-Dimensional Autonomous Systems and Chaos

4 3

2 z(t)O -I

-2 -3 -4

-3

-2

-I

y(t) 0 I

2

0 -1

3

-2 -3

-4

I x(t)

44

Figure 8.2: Phase portrait for system (8.2). You can rotate the figure in Maple using the left mouse button. Make a sketch of some trajectories in xyz space. Solution. The origin is the only critical point. Consider the transformations. Then

= (-3xt + lOyt)- 2(-2xt + 5yt) = Xt = x- 2y, y = -.Yt = -(-2xt + 5yt) = 2xt- 5yt = 2x + y, Z = -yl + z"t = -(-2Xt + 5yt) + (-2Xt + 2yt + 3zt) = 3(-Yt + Zt) =

.X= it- 2.Yt

3z.

System (8.3) is already in canonical form, and the eigenvalues are >.. 1.2 = 1±i and >..3 = 3; hence the critical point is hyperbolic. The system can be uncoupled; the critical point at the origin in the xy plane is an unstable focus. A phase plane portrait is given in Figure 8.3. There is an unstable manifold corresponding to the xy plane; any trajectory starting on this plane will tend to the origin as t ｾ＠ -oo. The z-axis is also an unstable manifold. lf z > 0, i; > 0, and if z < 0, < 0. Putting all of this together, the trajectories spiral away from the origin, as depicted in Figure 8.4.

z

Example 3. Solve the initial value problern

x=z-x.

y=

(8.4)

-y,

z=z-11x+16

123

8.1. Linear Systems and Canonical Forms

Figure 8.3: Some trajectories in the xy plane. This plane is a two-dimensional unstable manifold.

4 3 2 1 Z{t) 0

-4 -3 -2

-1

-2 -3 -4-4

-I

-3

-

2

I

-1 0 y(t)

Ox(t)

2 1 2

3

Figure 8.4: Phase portmit for system (8.3). You can rotate the figure in Maple using the left mouse button.

8. Three-Dimensional Autonomous Systems and Chaos

124 with x(O) space.

= y(O) = z(O) = 0.8, and plot the so1ution curve in three-dimensional

Solution. System (8.4) can be uncoupled. The differential equation y = -y has general solution y(t) = yoe-t, and substituting YO = 0.8 gives y(t) = 0.8e- 1 • Now z = x + x, and therefore the equation i. = z- 17x + 16 becomes (x +x) = (x +x) -17x + 16,

which simplifies to

x + l6x =

16.

Take Laplace transforms of both sides and insert the initial conditions to obtain

1

x(s)

=; -

0.2s + 16.

s2

Take inverse transforms to get x(t)

and therefore z(t)

= 1- 0.2cos(4t)

= 1 + 0.8 sin(4t) -

0.2 cos(4t).

The solution curve is plotted in Figure 8.5.

2 1.8 1.6

1.4

z(t) I

0.8 0.6 0.4

0.2

Ｐ

ｯＨｾＮｉＰＲＳＭＱ＠

I

.

. 0.4 0.5 0.6 0.7 I 2 y(t)

.

x(t)

Figure 8.5: The solution curve for the initial value problern in Example 3. The trajectory ends up on an ellipse in the y = 0 plane.

125

8.2. Nonlinear Systems and Stability

8.2

Nonlinear Systems and Stability

If the critical point of a three-dimensional autonomous system is hyperbolic, then the linearization methods of Hartman can be applied (see [3, Chapter 3]). If the critical point is not hyperbolic, then other methods need to be used.

Theorem 1. eonsider the differential equation

i = f(x),

xe

91",

where f e e 1(E) and Eis an open subset offlln containing the origin. Suppose that f(O) = 0 and that the Jacobian matrix has n eigenvalues with nonzeroreal part. Then in a small neighborhood of x = 0, there exist stable and unstable manifolds Ws and Wu with the same dimensions ns and nu as the stable and unstable manifolds (Es, Eu) of the linearized system

i= Jx, where Ws and Wu are tangent to Es and Eu at x = 0. A proof of this theorem can be found in Hartman's book (see [3, Chapter 3]). If the Jacobian matrix J has eigenvalues with zero real part, then there exists a so-called center manifold, which in generat is not unique.

Theorem 2 (The Center Manifold Theorem). Let f e er (E) (r 2:: 1), where Eis an open subset ojmn containing the origin. /ff(O) = 0 and the Jacobian matrix has ns eigenvalues with negative real part, nu eigenvalues with positive real part, and nc = n - ns - nu purely imaginary eigenvalues, then there exists an ne-dimensional center manifold Wc of class er that is tangent to the center manifold Ec of the linearized system. To find out more about center manifolds, see the book ofWiggins [6].

Example 4. Determine the stable, unstable, and center manifolds of the nonlinear system

y=-y, i

=

-2z.

Solution. There is a unique critical point at the origin. This system is easily solved, and it is not difficult to plot phase portraits for each of the uncoupled systems. The solutions are x(t) c/-r, y(t) e2e-r, and z(t) e- 2r. The eigenvalues and correspondingeigenvectorsoftheJacobianmatrixare.Ä. 1 = 0, (1, 0, O)T,.Ä.2 = -1, (0, 1, O)T, and .Ä.J -2, (0, 0, l)T. In this case, Wc Ec, the x-axis, and the yz plane forms a two-dimensional stable manifold, where Ws = Es. Note that the center manifold is unique in this case, but it is not in general.

=

=

=

=

=

126

8. Three-Dimensional Autonomous Systems and Chaos

Example 5. Solve the nonlinear differential system

X =-X, y=-y+x2,

= z + x2 ,

i;

and determine the stable and unstable manifolds. Solution. The point 0 = (0, 0, 0) is a unique critical point. Linearize by finding the Jacobian matrix. Hence aP

l=

(

ｾ＠

ax

aR

äX

where i

= P(x, y, z), y =

Q(x, y, z), and i; Jo

=

= R(x, y, z). Therefore,

( -1 0 0) 0 0

-1 0

0 1

,

and the origin is an unstab1e critica1 point. Note that two of the eigenvalues are negative. These give a two-dimensional stable manifold, which will now be defined. The differential equation i = -x is integrable and has solution x(t) = C 1e-r. The other two differential equations are linear and have solutions y(t) = C2e-r + C[(e-r - e- 2') and z(t) = C3e1 + ｾＨ･Ｇ＠ - e- 21 ). Now A+(x) = 0 if

c2

and only if c3 + ｾ＠ = 0, where XE m 3, Cl Therefore, the stable manifold is given by WS

= x(O), c2 = y(O), and c3 = z(O).

= {X E !R3: Z = - ｾ Ｒ＠ }.

Using similar arguments, A-(x) = 0 if and only if C1 = C2 = 0. Hence the unstable manifold is given by Wu = {x E !R3 :X = y = 0}.

Note that the surface Ws is tangent to the xy plane at the origin. Example 6. Sketch a phase portrait for the system

i = x + y - x(x 2 + y 2), y = -x (8.5)

i; =

-z.

+y -

y(x2 + y2),

8.2. Nonlinear Systems and Stability Solution. Convert to cylindrical polar coordinates by setting x y = r sin (). System (8.5) then becomes

127

= r cos 0 and

; = r(l- r2),

ö=

-1,

i=-z. The origin is the only critical point. The system uncouples; in the xy plane, the flow is clockwise and the origin is an unstable focus. If z > 0, then i < 0, and if z < 0, then i > 0. If r = I, then r = 0. Trajectories spiral towards the xy plane and onto the Iimit cycle, say r 1, of radius 1 centered at the origin; hence A+(x) = rl if X :/:: 0 and rl is a stable Iimit cycle. A phase portrait is shown in Figure 8.6.

3 2

z(t)O

-I -2

-3

Figure 8.6: Trajectories are attracted to a stable Iimit cycle in the xy plane. Lyapunov functions were introduced in Chapter 6 and were used to determine the stability of critical points for certain planar systems. The theory is easily extended to the three-dimensional case, as the following examples demonstrate. Once again, there is no systematic way to determine the Lyapunov functions, and they are given in the question. Example 7. Prove that the origin of the system

8. Three·Dimensional Autonomous Systems and Chaos

128

x=

y=

-2y+ yz, x(l- z),

z=xy

is stable but not asymptotically stable by using the Lyapunov function V (x, y, z) =

ax 2 + by 2 + cz 2. Solution. Now

dV oVdx oVdy avdz = - - + - - + - - =2(a -b+c)xyz+2(b-2a)xy. dt ax dr ay dt az dt

-

If b = 2a and a = c > 0, then V(x, y, z) > 0 for all :x =I= 0 and r1f, = 0. Thus the trajectories lie on the ellipsoids defined by x 2 + 2y 2 + z2 = r 2• The origin is thus stable but not asymptotically stable. E:xample 8. Prove that the origin of the system

x= - y -

xi + z2 - x3 ,

y = x +z3 - y3, z. = -xz - x 2 z - yz 2 - z5 is asymptotically stable by using the Lyapunov function V (x, y, z) = x 2 + y 2 + z2 • Solution. Now

dV dt

av dx ax dt

av dy ay dt

av dz az dt

4

4

2 2

2 2

6

- = - - + - - + - - = - 2 ( x +y +x z +x y +z ). Since r1f, < 0 for x, y, z =I= 0, the origin is asymptotically stable. In fact, the origin is globally asymptotically stable since A +{:x) = {0, 0, 0) for all :x e !R3 .

8.3

The Rössler System and Chaos

8.3.1 The Rössler Attractor. In 1976, Otto E. Rössler [10] constructed the following three-dimensional system of differential equations: x = -(y +z),

(8.6)

y=x+ay, z= b +xz- cz,

where a, b, and c are all constants. Note that the only nonlinear term appears in the i; equation and is quadratic. As the pararneters vary, this simple system can display

a wide range of behavior. Set a = b = 0.2, for exarnple, and vary the parameter c. The dynamics of the system can be investigated using the Maple package. Four examples are considered here. Transitional trajectories have been omitted to avoid confusion. The initial conditions are x(O) = y(O) = z(O) = 1 in all cases.

129

8.3. The Rössler System and Chaos

Definition 2. A Iimit cycle is of period one if x(t) = x(t + T) for some constant T called the period. The periodic orbit is called a period-one cycle. When c = 2.3, there is a period-one Iimit cycle that can be plotted in threedimensional space. Figure 8.7(a) shows the Iimit cycle in three-dimensional space, and the periodic behavior with respect to x(t) is shown in Figure 8.7(b).

4

.r(t)

2

0 I

-2

(a)

(b)

Figure8.7: (a)Alimitcycleforsystem(8.6)whenc = 2.3. (b)Period-onebehavior for x(t). When c = 3.3, there is period-two behavior. Figure 8.8(a) shows the closed orbit in three-dimensional space, and the periodic behavior is shown in Figure 8.8(b). Notice that there are two distinct amplitudes in Figure 8.8(b). This periodic behavior can be easily detected using ｐｯｩｮ｣｡ｲｾ＠ maps (see Chapter 9). 5.3, there is period-three behavior. Figure 8.9(a) shows the When c closed orbit in three-dimensional space, and the periodic behavior is shown in Figure 8.9(b). Note that there are three distinct amplitudes in Figure 8.9(b). When c = 6.3, the system displays what appears tobe random behavior. This type of behavior has been Iabeted deterministic chaos. A system is called deterministic if the behavior of the system is determined from time evolution equations and the initial conditions alone, as in the case of the Rössler system. Nondeterministic chaos arises when there are no underlying equations, as in the National Lottery, or there is noisy or random input. This text will be concerned with deterministic chaos only, and it will be referred to simply as chaos from now on.

=

8.3.11 Chaos. Chaos is a multifaceted phenomenon that is not easil y classified or identified. There is no universally accepted definition for chaos, but the following characteristics are nearly always displayed by the solutions of chaotic systems:

8. Three-Dimensional Autonomous Systems and Chao s

130

6

2

80

(a)

(b)

Figure 8.8: (a) A period-two Iimit cycle for system (8.6) when c = 3.3. (b) Periodtwo behavior for x(t). 10

x(t)

18

s

16 14

12 z(t) 10

8 6 4 2

0 I

-S

(a)

(b)

Figure 8.9: (a)Aperiod-three limitcycle for system (8.6) when c = 5.3. (b) Periodthree behavior for x(t). 1. long-term aperiodic (nonperiodic) behavior; 2. sensitivity to initial conditions;

3. fractal structure.

8.3. The Rössler System and Chaos

131

Consider each of these items independently. Note, however, that a chaotic system generally displays all three types of behavior listed above. Case 1. It is very difficult to distinguish between aperiodic behavior and periodic behavior with a very long period. For example, it is possible for a chaotic system to have a periodic Solution of period I oHJO! Case 2. A simple method used to test whether or not a system is chaotic is to check for sensitivity to initial conditions. Figure 8.lO(a) shows the trajectory in threedimensional space, and Figure 8.10(b) illustrates how the system is sensitive to the choice of initial conditions.

Definition 3. An attractor is a minimal closed invariant set that attracts nearby trajectories 1ying in the domain of stability (or basin ofattraction) onto it. Definition 4. Astrange attractor (chaotic attractor, fractal attractor) is an attractor that exhibits sensitivity to initial conditions. An example of a strange attractor is shown in Figure 8.lO(a). Another method for establishing whether or not a system is chaotic is to use the Lyapunov exponents (see Chapter 14). A system is chaotic if at least one of the Lyapunov exponents is positive (see [7], for example). This implies that two trajectories which start close to each other on the strange attractor will diverge as time increases, as depicted in Figure 8.1 O(b). 12 10

8 .l{l)6

4

2 0 -2 ｾ＠

-6 ｾ＠

-10

(a)

(b)

Figure 8.10: (a) The chaotic attractor for system (8.6) when c = 6.3. (b) Time series plot of x(t) showing sensitivity to initial conditions; the initial conditions for one time series are x(O) = y(O) = z(O) = 1 and for the other are x(O) = 1.01, y(O) = z(O) = 1.

132

8. Three-Dimensional Autonomous Systems and Chaos

One interesting feature of these strange attractors is that it is sometimes possible to reconstruct the attractor from time series data alone; see [1], for example.

Case 3. The solution curves to chaotic systems generally display fractal structure (see Chapter 18). The structure of the strange attractors for higher-dimensional systems may be complicated and difficult to observe clearly. To overcome these problems, Poincare maps, which exist in tower-dimensional spaces, can be used, as in Chapter 9.

8.4 The Lorenz Equations, Chua's Circuit, and the Belousov-Zhabotinski Reaction There are many examples of applications of three-dimensional autonomous systems to the real world. These systems obey the existence and uniqueness theorem from Chapter 1, but the dynamics can be much more complicated than in the twodimensional case. The following examples taken from meteorology, electric circuit theory, and chemical kinetics have been widely investigated in recent years. 8.4.1 The Lorenz Equations. In 1963, MIT meteorologist Edward Lorenz [ 11] constructed a highly simplified model of a convecting fluid. This simple model also displays a wide variety of behavior and for some parameter values is chaotic. The equations can be used to model convective flow up through the center and down on the sides of hexagonal columns. The system is given by

(8.7)

i = u(y- x), y = rx- y -xz, i: = xy- bz,

where x measures the rate of convective overturning, y measures the horizontal temperature variation, z measures the vertical temperature variation, CT is the Prandtl number, r is the Rayleigh number, and b is a scaling factor. The Prandtl number is related to the fluid viscosity, and the Rayleigh number is related to the temperature difference between the top and bottom of the column. Lorenz studied the system when CJ = 10 and b = ｾＮ＠ The system can be considered tobe a highly simplified model for the weather. Indeed, satellite photographs from space show hexagonal patterns on undisturbed desert floors. The astanishing conclusion derived by Lorenz is now widely labeled as the butterfly effect. Even this very simple model of the weather can display chaotic phenomena. Since the system is sensitive to initial conditions, small changes to wind speed (convective overturning), for example, generated by the flap of a butterflies wings, can change the outcome of the results considerably. For example, a butterfty flapping its wings in Britain could cause or prevent a hurricane from occurring in the Bahamas in the not-so-distant future. Of course, there are many more variables that should be considered when trying to model weather

8.4. Lorenz Equations, Chua's Circuit, Belousov-Zhabotinski Reaction 133 systems, and this simplified model illustrates some of the problems meteorologists have to deal with. Note that most nonlinear systems display steady-state behavior most of the time, so it is possible to predict, for example, the weather, the motion of the planets, the spread of an epidemic, the motion of a driven pendulum, or the beat of the human heart. However, nonlinear systems can also display chaotic behavior, where prediction becomes impossible. Some simple properties of the Lorenz equations will now be Iisted, and all of these characteristics can be investigated with the aid of the Maple package:

1. System (8.7) has natural symmetry (x, y, z)--. (-x, -y, z). 2. The z-axis is invariant. 3. The ftow is volume contracting since divx = -(u is the vector field.

+ b + 1)

< 0, where x

4. lf 0 < r < 1, the origin is the only critical point, and it is a global attractor.

5. At r

= 1, there is a bifurcation, and there are two more critical points at C 1 =

(.Jb(r- 1), ,Jb(r- 1), r - 1) and r - 1).

c2 =

(-,Jb(r- 1), -,Jb(r- 1),

6. At r = rH ｾ＠ 13.93, there is a homoclinic bifurcation (see Chapter 10), and the system enters a state of transient chaos. 7. At r ｾ＠

24.06, a strange attractor is formed.

8. lf 1 < r < ro. where ro ｾ＠ both stable.

24.74, the origin is unstable and C1 and C2 are

9. At r > r o, C J, and C2 lose their stability by absorbing an unstable Iimit cycJe in a subcritical Hopf bifurcation. For more details, see the work of Sparrow [9] or most textbooks on nonlinear dynamics. Most of the results above can be observed by plotting phase portraits or time series using the Maple package. A strange attractor is shown in Figure 8.11. The trajectories wind around the two critical points C 1and C2 in an apparently random unpredictable manner. The strange attractor has the following properties: • The trajectory is aperiodic (or not periodic). • The trajectory remains on the attractor forever (the attractor is invariant). • The general form is independent of initial conditions. • The sequence of windings is sensitive to initial conditions. • The attractor has fractal structure.

134

8. Three-Dimensional Autonomous Systems and Chaos 44 ·

42 40 38 36 34 32

30 z(t)

28 26

24 22

20 -

18

16 14 12

10 8

6

Ｑ ＶｦｩＲＸ

Ｘ ｉ ＶＴ ｾ Ｒ ｾ ｟ Ｒ ｾ x(rJ

ｾ Ｌ ｾＭﾭ

- 12 -18

Figure 8.11 : Astrange attractor for the Lorenz system when u = 10, b = r = 28.

J, and

A variation on the Lorenz model has recently been discovered by Guanrong Chen and Tetsushi Ueta. The equations are

x = u(y- x) ,

y = (r- u)x + ry- xz, z=xy -bz.

(8.8)

8.4.11 Chua 's Circuit. Elementary electric circuit theory was introduced in Chapter 1. In the mid-l980s, Chua modeled a circuit that was a simple oscillator exhibiting a variety of bifurcation and chaotic phenomena. The circuit diagram is given in Figure 8.13. The circuit equations are given by dv1 l = - (G(v2- VI)- f(vl)), dt Cl dv2 1 dt = C2 (G(vl - V2) + i),

-

(8.9)

di dt =

V2

-z;·

where v1, v2, and i are the voltages across C 1 and C2 and the current through L, respectively. The characteristic of the nonlinear resistor N R is given by f(vJ) = GbVJ

+ 0.5(Ga- Gb) (lv1 + Bpl -lvl

- Bpl),

8.4. Lorenz Equations, Chua's Circuit; Belousov-Zhabotinski Reaction 135

40 30

z(t)

20 10

Figure 8.12: Astrange attractor for system (8.8) when u

= 35, b = 3, and r = 28.

R

L

where G

=

20.8mB, R

*·=

Bp = 1.85V.

Figure 8.13: Chua's electric circuit. 'f'ypical parameters used are C1 = lO.lnF, C2 = lOlnF, L = 14200, r = 63.80, Ga = -0.865mS , Gb = -0.519mS , and

In the simple case, Chua's equations can be written in the following dimensionless form:

x = a(y- x- g(x)), y = x- y +z, z = -by,

(8.10)

where a and b are dimensionless parameters. The function g(x) has the form g(x) = cx

where c and d are constants.

1

+ 2(d- c)(lx + 11-lx- 11),

8. Three-Dimensional Autonomous Systems and Chaos

136

Chua's circuit is investigated in some detail in [5] and exhibits many interesting phenomena including period-doubling cascades to chaos, intermittency routes to chaos, and quasiperiodic routes to chaos, for example. All of these phenomena wiii be discussed in later chapters. For certain parameter values, the solutions lie on a double-scroll attractor, as shown in Figure 8.14.

3· 2 Z(l} 0 · -1 ·

-2 -

-3 -0.4

-2 -0.2

Figure 8.14: Chua's double-scroll attractor: Phaseportrait for system (8.10) when a = 15, b = 25.58, c = _,,andd = ＭｾＮ＠ Theinitialconditionsarex(O) = -1.6 y(O) 0, and z(O) 1.6.

=

=

The dynamics are more complicated than those appearing in either the Rössler or Lorenz attractors. Chua's circuit has proved tobe a very suitab1e subject forstudy since 1aboratory experiments produce results that match very weil with the results of the mathematical model.

8.4.111 The Belousov-Zhabotinski Reaction. Periodic chemical reactions such as the Landolt clock and the Belousov-Zhabotinski reaction provide wonderful examples of relaxation osciiiations in science (see [2, 3, 7]). They are often demonstrated in chemistry classes or used to astound the public at university open days. The first experiment was conducted by the Russian bioehernist Boris Belousov in the 1950s, and the results were not confinned until as late as 1968 by Zhabotinski. Consider the following recipe for a Belousov periodic chemical reaction.

Ingredients. Solution A: Malonic acid, 15.6 gmll. Solution B: Potassium bromate, 41.75 gm/1, and potassium bromide, 0.006 gmll.

8.4. Lorenz Equations, Chua's Circuit, Belousov-Zhabotinski Reaction 137 Solution C: Cerium IV sulphate, 3.23 gm/1, in 6M sulphuric acid. Solution 0: Ferroin indicator.

Procedure. Add 20 mls of solution A and 10 mls of solution B to a mixture of l 0 mls of solution C and I ml of solution 0. Stir continuously at room temperature. The mixture remains blue for about ten minutes and then begins to oscillate blue-greenpink and back again with a period of approximately two minutes. This reaction is used by the Chemistry Oepartment at Manchester Metropolitan University during open days and is always a popular attraction. Following the methods ofField and Noyes (see [3]), the chemical rate equations for an oscillating Belousov-Zhabotinski reaction are frequently written as BrO:J + Br--+ HBr02 + HOBr, HBr02 + Br- -+ 2HOBr, BrO:J + HBr02 -+ 2HBr02 + 2Mox.

Rate= kJ[BrO:J][Br-], Rate = k2[HBr02][Br-], Rate= kJ[BrO:J][HBr02],

2HBr02-+ BrO:J + HOBr, (8.11)

OS+ Mox -+

1

2csr-,

Rate = ks[OS][Mox].

where OS represents all oxidizable organic species and Cis a constant. Note that in the third equation, species HBr02 stimulates its own production, a process called autocatalysis. The reaction rate equations for the concentrations of intermediate species x = [HBr02], y = [Br-]. and z = [Mox] are

x = k1ay- k2xy + k3ax- 2k4x 2, 1 k . y = - 1ay- k2xy + 2Cksbz, (8.12)

z=

2k3ax - ksbz,

where a = [BrO:J] and b = [OS] are assumed tobe constant. Taking the transformations X= 2k4X. r = ksbt, ksa system (8.12) becomes

(8.13)

q Y - XY + X (1 - X) dX -= EI dr -qY-XY+CZ dY -= €2 dr dZ dr =X- Z,

8. Tbree-Dimensional Autonomous Systems and Chaos

138

4ab, and q = 2kk 'k4 • Next, one assumes that E2 « 1 so where Et = hl, E2 = 'f'{' k3a 2 3 2 3 that ｾ＠ is large unless the numerator -q Y - X Y + C Z is also small. Assurne that

y

= Y* = ...!:..!_ q+X

at all times, so the bromide concentration Y = [Br-] is in a steady state compared to X. In this way, a three-dimensional system of differential equations is reduced to a two-dimensional system of autonomous ODEs E/X =X(l-X)- X-qCZ,

dr

X+q

dZ -=X-Z.

(8.14)

dr

For certain parameter values, system (8.14) has a Iimit cycle that represents an oscillating Belousov-Zhabotinski chemical reaction as in Figure 8.15. Example 9. Find and classify the critical points of system (8.14) when Et = 0.05, q = 0.01, and C = I. Plot a phase portrait in the first quadrant. Solution. There are two critical points, one at the origin and the other at A (0.1365, 0.1365). The Jacobian matrix is given by J =

( flI (t -

2

X- ｾＫｱ＠

z

+ (X-q)f) (X+q)

flI

ｾ＠

(CLJ!.) X+q ) . -1

It is not difficult to show that the origin is a saddle point and A is an unstable node. The direction vectors can be plotted using the Maple package. A phase portrait showing periodic oscillations is given below. The period of the Iimit cycle in Figure 8.15 is approximately 3.4. The trajectory moves quickly along the right and left branches of the Iimit cycle (up and down) and moves relatively slowly in the horizontal direction. This accounts for the rapid color changes and time spans between these changes. It is important to note that chernical reactions are distinct frorn rnany other types of dynamical systems in that closed chernical reactions cannot oscillate about their chemical equilibrium state. This problern is easily surmounted by exchanging mass or introducing a ftow with the chemical reaction and its surroundings. For exarnple, the Belousov-Zhabotinski reaction used during university open days is stirred constantly and mass is exchanged. lt is also possible for the BelousovZhabotinski reaction to display chaotic phenornena; see [2), for example. Multistahle and bistable chemical reactions are also discussed in [3]. In these cases, there is an inftow and outftow of certain species and rnore than one steady state can coexist.

139

8.5. Maple Commands

0.4

X 0.6

0.8

Figure 8.15: A Iimit cycle in the XZ plane for system (8.14) when EI = 0.05, q=0.01,andC=l.

8.5

Maple Commands

The Maple commands below may be edited to produce solutions and diagrams for all of the examples and exercises appearing in Chapter 8. >with(linalg): >A:=matrix([[l,0,-4], [0,5,4), [-4,4,3)));

>eigenvects (A) ;

[3, I, [-2, -2, 1]],

[-3, I, [-2, 1, -2]],

[9, 1, [1, -2, -2]]

># See Exercise 1. >

>with(DEtools); >a:=0.2;b:=0.2;c:=2.3; >Rossler:=diff(x(t),t)=-y-z,diff(y(t),t)=x+a*y, >diff(z(t),t)=b+x*z-c*z; >DEplot3d({Rossler},{x(t),y(t),z(t)},t=50 .. 120, >[[x(O)=l,y(O)=l,z(O)=l]], >scene=[x(t),y(t),z(t)],stepsize=0.05); >

># See Figure 8.6(a).

140

8. Three-Dimensional Autonomous Systems and Chaos

>

>DEplot ( {Rossler}, {x (t), y ( t), z (t)}, t=50 .. 120, [ [x (0) =1, y(O) =1, >z(0)=1J], >scene=[t,x(t)],stepsize=0.05); >

># See Figure 8.6(b). >

>a:=15.6;b:=25.58;c:=-5/7;d:=-8/7; >Chua:=diff(x(t),t)=a*(y-x-(c*x+0.5*(d-c)*(abs(x+1) -abs(x-1)))), >diff(y(t),t)=x-y+z, >diff(z(t),t)=-b*y; >

>DEplot3d( {Chua}, {x(t), y (t), z (t)}, t=O .. 40, [ [x (0) =1. 6,y(0) =0, >z(0)=-1.6], >[x(0)=-1.6,y(0)=0,z(0)=1.6]],scene=[x(t),y(t),z(t)] ,stepsize=0.05); >

>#See Figure 8.13.

8.6

Exercises

I. Find the eigenvalues and eigenvectors of the matrix A = (

ｾ＠ ｾ＠

-4 4

Hence show that the system

3

oｾ n

x=

J=

ｾＴＩＮ＠

Ax can be transformed into

u=

Ju, where

Ｓ＠

Sketch a phase portrait for the system u = Ju.

2. Classify the critical point at the origin for the system

i = x + 2z,

.Y = y - 3z,

z= 2y + z.

3. Find and classify the critical points of the system

i=x-y,

y=y+i.

z=x-z.

4. Consider the system

i=-x+(J..-x)y,

y=

x- (>..- x)y- y + 2z,

.

y 2

z = - - z.

where).. =::: 0 is a constant. Show that the first quadrant is positively invariant and that the plane x + y + 2z = constant is invariant. Find ).. +(p) for p in the first quadrant given that there are no periodic orbits there.

8.6. Exercises

141

5. Prove that the origin of the system 3 x. = -x-y 2+ xz-x,

y.

= -y + z2+ xy-y 3 ,

z. = -z+x 2+ yz-z 3

is globally asymptotically stable. 6. Determine the domain of stability for the system

x= -ax + xyz,

y = -by + xyz,

z= -cz + xyz.

7. A three-dimensional Lotka-Volterra model is given by

x=

x(l-2x+y-5z).

y=

y(l-5x-2y-z).

z=

z(l+x-3y-2z).

f.i, f4, fi). Plot

Prove that there is a critical point in the first quadrant at P ( possible trajectories and show that there is a solution plane x Interpret the results in terms of species behavior.

+y +z=

ｾＮ＠

8. Using the Maple package, investigate (i) the Rössl er system (Section 8.3.1) for varying values ofthe parameter c;

(ii) the Lorenz system (Section 8.4.1) for varying values of the parameter r;

(iii) the Chua system (Section 8.4.II) as the parameters a, b, c, and d are varied. 9. Plot some time series data for the Lorenz system (8.7) when a = 10, b = ｾＮ＠ and 166 :::: r :::: 167. When r = 166.2, the solution shows intermittent behavior; there are occasional chaotic bursts between what Iooks like periodic behavior. 10. Consider system (8.14) given in the text to model the periodic behavior of the Belousov-Zhabotinski reaction. By considering the isoclines and gradients of the vector fields, explain what happens to the solution curves for EI I and appropriate values of q and C.

«

Recommended Reading [1] M. Zoltowski, An adaptive reconstruction of chaotic attractors out of their single trajectories, Signal Process., 80-6 (2000), 1099-1113. [2] N. Arnold, Chemical Chaos, Hippo, London, 1997. [3] S. K. Scott, Oscillations, Waves, and Chaos in Chemical Kinetics, Oxford Science Publications, Oxford, 1994. [4] R. C. Hilborn, Chaos and Nonlinear Dynamics: An lntroductionfor Seienrists and Engineers, Oxford University Press, Oxford, 1994.

8. Three-Dimensional Autonomous Systems and Chaos

142

[5] L. P. Shil'nikov, Chua's circuit: Rigorous results and future prob1ems,lntermat. J. Bifurcation Chaos, 4 (1994), 489-519.

[6] S. Wiggins,lntroduction to Applied Nonlinear Dynamical Systemsand Chaos, Springer-Verlag, Berlin, New York, Heidelberg, 1990. [7] J. P. Eckmann, S. 0. Kamphorst, D. Ruelle, and S. Ciliberto, Liapunov exponents from time series, Phys. Rev. A, 34-6 (1986), 4971-4979.

[8] R. Field and M. Burger, eds., Oscillations and Travelling Waves in Chemical Systems, Wiley, New York, 1985. [9] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer-Verlag, New York, 1982.

[10] 0. E. Rössler, An equation for continuous chaos, Phys. Lett., 57A (1976), 397-398. [11] E. N. Lorenz, Deterministic non-periodie flow, J. Atmos. Sei., 20 (1963), 130-141.

9 Poincare Maps and Nonautonomaus Systems in the Plane

Aims and Objectives • To introduce the theory of Poincare maps. • To compare periodic and quasiperiodic behavior. • To introduce Hamiltonian systems with two degrees of freedom. • To use Poincare maps to investigate a nonautonomaus system of differential equations. On completion of this chapter, the reader should be able to • understand the basic theory of Poincare maps; • plot return maps for certain systems; • use the Poincare map as a tool for studying stability and bifurcations. Poincare maps are introduced viaexample using two-dimensional autonomous systems of differential equations. They are used extensively to transform complicated behavior in the phase space to discrete maps in a tower-dimensional space. Unfortunately, this nearly always results in numerical work since analytic solutions can rarely be found. The Poincare commands within the DEtools package are used to plot Poincare maps for Hamiltonian systems with two degrees of freedom. A periodically forced

144

9. Poincare Maps and Nonautonomous Systems in the Plane

nonautonomaus system of differential equations is introduced, and Poincare maps are used to determine stability and plot bifurcation diagrams. Discrete maps are discussed in more detail in later chapters of the book.

9.1

Poincare Maps

When plotting the solutions to some nonlinear problems, the phase space can become overcrowded and the underlying structure may become obscured. To overcome these difficulties, a basic tool was proposed by Henri Poincare [5] at the end of the 19th century. As a simple introduction to the theory of Poincare (or first return) maps, consider two-dimensional autonomous systems of the form

x= (9.1)

y=

P(x, y),

Q(x, y).

Suppose that there is a curve or straight line segment, say E, that is crossed transversely (no trajectories aretangential to E). Then E is called a Poincare section. Consider a point ro, lying on E. As shown in Figure 9.1, follow the flow of the trajectory until it next meets E at a point r1. This point is known as the first return of the discrete Poincare map P : E --+> E, defined by 'n+l = P(rn),

where r n maps to r n+ 1 and all points lie on E. Finding the function Pis equivalent to solving the differential equations (9.1). Unfortunately, this is very seldom possible, and one must rely on numerical solvers to make any progress.

Poincare section

Figure 9.1: A first return on a Poincare section. Definition 1. A point r* that satisfies the equation P(r*) = r• is called a fixed point ofperiod one. To illustrate the method for finding Poincare maps, consider the following two simple examples (Examples 1 and 2), for which P may be determined explicitly. Example 1. Byconsideringthelinesegment E 0}, find the Poincare map for the system

= {(x, y) e m2 : 0:::: X::::

x=-y-xJx2+y2, (9.2)

y = x - yJx2 + y2

and Iist the first eight returns on E given that ro = 1.

1, y

=

145

9.1. Poincare Maps Solution. Convert to polar coordinates. System (9.2) then becomes

r=

-r 2 ,

(9.3) The origin is a stable focus and the ftow is counterclockwise. A phase portrait showing the solution curve for this system is given in Figure 9.2.

Figure 9.2: A trajectory starting at (1, 0), (0 :S t :S 40) for system (9.3). The set of equations (9.3) can be solved using initial conditions r(O) = 1 and 0(0) = 0. The solutions are given by I =, 1+ t

r(t)

O(t)

= t.

Trajectories flow around the origin with a period of 21r. Substituting for t, the flow is defined by I r(t) = 1 + O(t) ·

The ftow is counterclockwise, and the required successive returns occur when 0 = 2rr, 4rr, .... A map defining these points is given by I

rn=---

1 + 2nrr on :E, where n = 1, 2, .... As n --+- oo, the sequence of points moves towards the fixed point at the origin as expected. Now

rn+ ｾＭ

1

-

1 + 2(n + I)Jr ·

Elementary algebra is used to determine the Poincare return map P, which may be expressed as

146

9. Poincare Maps and Nonautonomous Systems in the Plane

The first eight returns on the line segment E occur at the points ro = I, = 0.13730, r2 = 0.07371, r3 = 0.05038, r4 = 0.03827, rs = 0.03085, r6 = 0.02584, r1 = 0.02223, and rs = 0.01951 to five decimal places, respectively. r1

Check these results for yourself using the Maple program at the end of the chapter. Example 2. Use a one-dimensional map on the line segment E = {(x, y) e 9l 2 : 0 ::: x < oo, y = 0} to determine the stability of the Iimit cycle in the following system:

x = - y+ x ( 1 - ｾ＠ (9.4)

y= x +y ( 1-

x2 + y2) ,

Jx + 2

y 2) •

Solution. Convert to polar coordinates. Then system (9.4) becomes

; = r(l- r), (9.5)

B=L

The origin is an unstab1e focus, and there is a Iimit cycle, say r, of radius 1 centered at the origin. A phase portrait showing two trajectories is given in Figure 9.3. System (9.5) can be solved since both differential equations are separable. The solutions are given by 1

r(t) = 1 +Ce-''

6(t) = t

+ 60 ,

where C and 6o are constants. Trajectories flow around the origin with a period of 27r.

Figure 9.3: Two trajectories for system (9.5), one starting at {2, 0) and the other at (0.01, 0).

147

9.1. Poincare Maps Suppose that a trajectory startsoutside are then given by 1 r(t) = , 1 1- ie-t

r on :E, say at ro = 2. The solutions O(t) = t.

Therefore, a retum map can be expressed as fn

= 1-

1

2e-2mr

,

where n is a natural number. If, however, a trajectory starts inside r at, say, ro then r(t)

= 1 + e-1 ,

O(t)

= !,

= t,

and a retum map is given by fn

= ---:::-1 + e-2mr.

In both cases, rn ｾ＠ I as n ｾ＠ oo. The Iimit cycle is stable on both sides, and the Iimit cycle r is hyperbolic stable since rn ｾ＠ I as n ｾ＠ oo for any initial point apart from the origin. The next theorem gives a better method for determining the stability of a Iimit cycle.

Theorem 1. Define the characteristic multiplier M to be dPI M--

dr r•'

where r* is a fixed point of the Poincare map P corresponding to a Limit cycle, say r. Then if

1. IMI < I,

r

is a hyperbolic stable Limit cycle;

2. IMI > I,

r

is a hyperbolic unstable Iimit cycle;

3. IM I = I, and ｾ＠

"# 0, then the Limit cycle is stable on one side and unstable on the other; in this case r is called a semistable Iimit cycle.

Definition 2. A fixed point of period one, say r*, of a Poincare map P is called hyperbolic if IMI "# I. Theorem l is sometimes referred to as the derivative ofthe Poincare map test.

Example 3. Use Theorem 1 to determine the stability of the Iimit cycle in Example 2.

Solution. Consider system (9.5). The retum map along :E is given by

148

9. Poincare Maps and Nonautonomaus Systems in the Plane

(9.6)

=

Tn

ＭＺｾ＠

I+ ce-2mr'

where C is a constant. Therefore, (9.7)

'n+l

= I

+ Ce-2

> if kk=2 then > seqn:=so1ve({z[O],z[1],z[2],z[3]},{uu[3,0],uu[3,l],u u[3,2], > uu [ 3, 3])) : > > >

> >

> > > > >

elif kk=3 then seqn: =so1ve ( {z [ 0], z [ 1], z [2], uu [4, 2], z [3], z [4]}, {uu [4, 0], uu [4, 1], uu[4,2],uu[4,3],uu[4,4],ETA[4]}): elif kk=4 then seqn: =solve ( {z [ 0 l , z [ 1] , z [ 2], z [ 3] , z [ 4] , z [ 5] } , {uu [ 5, 0 l , uu [ 5, 1], uu [ 5, 2] • uu [ 5. 3] • uu [ 5, 4] , uu [ 5, 5] } ) : elif kk=5 then seqn: =sol ve ( { z [ 0 J , z [ 1] , z [ 2] , uu [ 6, 2] +uu [ 6, 4] , z [ 3] , z [ 4] , z [ 5] , z [ 6] } , {uu[6,0],uu[6,1],uu[6,2],uu[6,3],uu[6,4],uu[6,5],uu[6 ,6],ETA[6]}): fi:

>

> assign(seqn) :x:='x':i:='i':

vv[kk+1] :=sum(uu[kk+1,i]*x'(kk+1-i)*y'i,i=O .. kk+1): vx[kk+1] :=diff(vv[kk+1],x):vy[kk+1] :=diff(vv[kk+1],y): > ETA[kk+l] :=numer(ETA[kk+l]):od:

> > >

lprint('ETA4:=',ETA(4]) :lprint('ETA6:=',ETA[6]):

ETA4:= -3*a3+2*b2*a2 ETA6:= 372*a2'2*a3-248*a2'3*b2+30*b2'2*a3-20*b2'3*a2+80*b2* a4-9*a3*b3 -34*b3*b2*a2-60*a5+24*b4*a2

179

10.5. Exercises

The algorithm for computing the focal values (the ETA[kk+l] above) is given in detail in Section 10.1. Set a3 := 2 * a2 * b2/3 : and checkthat ETA6 does reduce to L(2) as given in the notes. The program will have to be extended to determine more eta values. For example, to determine ETAS, kend must be set to 7 and the simultaneaus equations for kk = 6 and kk = 7 have to be inserted into the program.

10.5

Exercises

1. Prove that the origin of the system

x = y- F(G(x)),

y=

is a center using the transformation u 2 argument.

- G'(x) H(G(x)) 2

= G(x) and the classical symmetry

2. Write a program to compute the first seven Lyapunov quantities of the Lienard system (10.8)

3. Using the results from question 2, prove that at most six small-amplitude Iimit cycles can be bifurcated from the origin of system (10.8). 4. Consider the system

x = y- (alx + a3x 3 + · ·· + a2n+lx2n+l),

y = -x.

Prove by induction that at most n small-amplitude Iimit cycles can be bifurcated from the origin. 5. Write a program to compute the first five Lyapunov quantities for the Lienard system

+ a2x 2 + · ·· + a1x 7) , y =-(X+ b2x 2 + b3x 3 + •· · + b6x 6).

x.

=y-

( a1x

6. Using the results from question 5, prove that ii (4, 2) = 2, ii (1, 2) = 4, and H(3, 6) = 4. Note that in H(u, v), u is the degree of Fand v is the degree of g.

7. Consider the generalized mixed Rayleigh-Lienard oscillator equations given by

x = y, Y = -x- a1y- b3ox 3 -

b21x 2y- b41x 4y- b03y 3 .

Prove that at most three small-amplitude Iimit cycles can be bifurcated from the origin.

10. Local and Global Bifurcations

180 8. Plot a phase portrait for the system .

.i = y,

2

y=x+x.

Determine an equation for the curve on which the homoclinic 1oop lies. 9. Consider the Lienard system given by

X= Y-

E(Q)X

+ a2x 2 + a3x 3), y =-X.

Prove that for sufficiently small asymptotic to a circle of radius

E,

there is at most one Iimit cycle that is

r= 10. Using the Maple package and DEplot, investigate the system .X

= y, y = x- x 3 + E(.l..y + x 2 y)

when E = 0.1 for values of .l.. from -1 to -0.5. How many Iimit cycles are there at most?

Recommended Reading [1] T. R. Blows and L. M. Perko, Bifurcation of Iimit cycles from centres and separatrix cycles of p1anar ana1ytic systems, SIAM Rev., 36 (1994), 341-376. [2] N. G Lloyd and S. Lynch, Small-amplitude Iimit cycles of certain Lienard systems, Proc. Roy. Soc. London Ser. A, 418 (1988), 199-208. [3] N. G. Lloyd, Limit cycles of polynomial systems, in T. Bedford and J. Swift, eds., New Directions in Dynamical Systems, London Mathematical Society Lecture Notes Series 127, Cambridge University Press, Cambridge, UK, 1988.

11 The Second Part of David Hilbert's

Sixteenth Problem

Aims and Objectives • To describe the second part of Hilbert's sixteenth problern. • To review the main results on the nurnber oflimit cycles of planar polynomial systems. • To consider the ftow at infinity after Poincare compactification. On completion of this chapter, the reader should be able to • state the second part of Hilbert's sixteenth problern; • describe the main results for this problem; • cornpactify the plane and construct a global phase portrait that shows the behavior at infinity for some simple systerns. The second part of Hilbert's sixteenth problern is stated and the rnain results are listed. To understand these results, it is necessary to introduce Poincare cornpactification, where the plane is rnapped on to a sphere and the behavior on the equator of the sphere represents the behavior at infinity for planar systems.

11.1

Statement of Problem and Main Results

Poincare began investigating isolated periodic cycles of planar polynornial vector fields in the 1880s. However, the generalproblern of determining the maxirnurn

182

11. The Second Part of David Hilbert's Sixteenth Problem

number and relative configurations of Iimit cycles in the plane has remained unresolved for over a century. Recall that Iimit cycles in the plane can correspond to steady-state behavior for a physical system (see Chapter 7), so it is important to know how many possible steady states there are. In 1900, David Hilbert presented a Iist of 23 problems to the International Congress of Mathematicians in Paris. Most of the problems have been solved, either completely or partially. However, the second part of the sixteenth problern remains unsolved. The Second Part of Hilbert's Sixteenth Problem. Consider planar polynomial systems of the form

i = P(x,y),

y=

(11.1)

Q(x,y),

where P and Q are polynomials in x and y. The question is to estimate the maximal number and relative positions of the Iimit cycles of system (11.1 ). Let Hn denote the maximum possible number of Iimit cycles that system (11.1) can have when P and Q are of degree n. More formally, the Hilbert numbers Hn are given by

Hn = sup{n(P, Q): öP, öQ ｾ＠

n},

where ö represents "the degree of" and n(P, Q) is the number oflimit cycles of system (11.1 ).

s

Dulac Theorem states that a given polynomial system cannot have infinitely many Iimit cycles. This theorem has only recently been proved independently by Ecalle et al. [5] and ll'yashenko [4], respectively. Unfortunately, this does not imply that the Hilbert numbers are finite. Of the many attempts to make progress in this question, one of the more fruitful approaches has been to create vector fields with as many isolated periodic orbits as possible using both local and global bifurcations. There are relatively few results in the case of generat polynomial systems even when considering local bifurcations. Bautin [9] proved that no more than three small-amplitude Iimit cycles could bifurcate from a critical point for a quadratic system. For a homogeneous cubic system (no quadratic terms), Sibirskii [8] proved that no more than five small-amplitude Iimit cycles could be bifurcated from one critical point. Recently, Zoladek [2] found an example where 11 Iimit cycles could be bifurcated from the origin of a cubic system, but he was unable to prove that this was the maximum possible number. Although easily stated, Hilbert's sixteenth problern remains almost completely unsolved. For quadratic systems, Shi Songling [7] has obtained a lower bound for the Hilbert number Hz ::: 4. A possible global phase portrait is given in Figure 11.1. The line at infinity is included and the properlies on this line are

183

11.2. Poincare Compactification

determined using Poincare compactification, which is described in Section 11.2. There are three small-amplitude Iimit cycles around the ongin and at least one other surrounding another critical point. Some of the pararneters used in this example are very small.

um;t cydo() Ｎｾ＠ ｾ＠

@

Small-amplitude Iimit cycles

Figure 11.1: A possible configuration for a quadratic system with four Iimit cycles: one of large arnplitude and three of small arnplitude. Blows and Rousseau [3] consider the bifurcation at infinity for polynomial vector fields and give examples of cubic systems having the following configurations: {(4), 1}, {(3), 2}, {(2), 5}, {(4), 2}, {(1), 5}, and {(2), 4}, where {(I), L} denotes the configuration of a vector field with l small-arnplitude Iimit cycles bifurcated from a point in the plane and L large-arnplitude Iimit cycles simultaneously bifurcated from infinity. There are many other configurations possible, some involving other critical points in the finite part of the plane, as shown in Figure 11.2. Recall that a Iimit cycle must contain at least one critical point. By considering cubic polynomial vector fields, in 1985, Li Jibin and Li Chunfu [6] produced an example showing that H3 ｾ＠ 11 by bifurcating Iimit cycles out of homoclinic and beterocHnie orbits; see Figure 11.2. Returning to the generat problem, in 1995, Christopher and Lloyd [1] considered the rate of growth of Hn as n increases. They showed that Hn grows at least as rapidly as n 2 log n. In recent years, the focus of research in this area has been directed at a small nurober of classes of systems. Perhaps the most fruitful has been the Lienard system, which is discussed in some detail in Chapter 12.

11.2 Poincare Compactification The method of compactification was introduced by Henri Poincare at the end of the 19th century. By making a simple transformation, it is possible to map

11. The Second Part of David Hilbert's Sixteenth Problem

184

ＯｾＭ -

Ｍｾ＠

---- ----

( ( ( ｾＩ｟＠

--',

·.,

\

Q ) \ \\ \--.:=/ /

\ I

', .:::__) '

.-/'

\ ----------I ＯｾＭＮ＠ ((Q) (QJ)) ＢＧｾＭ｟ＮＬ＠

,I

Figure 11.2: A possible configuration for a cubic system with 11 Iimit cycles. the phase plane onto a sphere. Note that the plane can be mapped to both the upper and lower hemispheres. In this way, the points at infinity are transfonned into the points on the equator of the sphere. Suppose that a point (x, y) in the plane is mapped to a point (X, Y, Z) on the upper hemisphere of a sphere, say S 2 ={(X, Y, Z) E !R 3 : X 2 + Y 2 + Z 2 = 1}. (Note that it is also possible to map onto the lower hemisphere.) The equations defining (X, Y, Z) in tenns of (x, y) are given by 1 y X ' ' Z= ' Y= X=

.Ji+r2

.Ji+r2

.Ji+r2

where r 2 = x 2 + y 2 . A centrat projection is illustrated in Figure 11.3. Consider the autonomous system ( 11.1 ). Convert to polar coordinates. Thus system (11.1) transfonns to

r = ,n fn+l (8) + ,n-l fn-l (8) + + ft (8), 0

(11.2)

0=

,n-l 8n+l (8)

•

0

+ ,n-28n-l8 +···+,-I 81 (8),

where fm and 8m are polynomials of degree m in cos 8 and sin 8. and system (11.2) becomes Hence p = -l.r, Let p = !. ｲｾ＠ r

P= 0=

-Pfn+l (8) 8n+1(8)

+ 0(p 2),

+ O(p).

Theorem 1. The critical points at infinity are found by solving the equations p =

e= 0 on p = 0, which is equivalent to solving

8n+l (8) = cos8Qn(cos8, sin 8)- sin 8Pn(cos8, sin8) = 0,

where Pn and Qn arehomogeneaus polynomials ofdegree n. Note that the solutions are given by the pairs 8; and 8; + rr. As long as 8n+ 1(8) is nonzero, there are n + 1 pairs of roots, and the jlow is clockwise when 8n+ 1(8) < 0 and counterclockwise when 8n+l (8) > 0.

185

11.2. Poincare Compactification

(x,y)

Figure 11.3: A mapping of (x, y) in the plane onto (X, Y, Z) on the upper part of the sphere. Todetermine the ftow near the critical points at infinity, one must project the hemisphere with X > 0 onto the plane X 1 with axes y and z or project the hemisphere with Y > 0 onto the plane Y 1 with axes x and z. The projection of the sphere S2 onto these planes is depicted in Figure 11.4. If n is odd, the antinodal points on S2 are qualitatively equivalent. If n is even, the antinodal points are qualitatively equivalent but the direction of the ftow is reversed. The ftow near a critical point at infinity can be determined using the following theorem.

=

=

Theorem 2. The flow defined on the yz plane (X = ±1), except at the points (0, ±I, 0), is qualitatively equivalent to the jlow defined by •

±y

= yzn p

(1z' zY) - (1z' zY) ' n

z Q

±z = z"+l P ＨｾＮ＠ ｾＩＮ＠ where the direction oftheflow is determinedfrom gn+l (0). In a similar way, the flow defined on the x z plane ( Y = ± 1), except at the points (±1, 0, 0), is qualitatively equivalent to theflow defined by •

n

±x = xz Q

(Xz' z I) - zn p (Xz' z1) .

11. The Second Part of David Hilbert's Sixteenth Problem

186

z

X Figure 11.4: The projections used to detennine the behavior at infinity.

I)

. n+IQ(x ± z=z -,- ,

z z

where the direction of the flow is determined from gn+l (0).

Example 1. Construct global phase portraits, including the ftow at infinity, for the following linear systems: (a)

.i = -x + 2y, y =

(b)

.i

2x

+ 2y;

= x + y, y = -x + y.

Solutions. (a) The origin is a saddle point (or col) with eigenvalues and corresponding eigenvectors given by >..1 = 3, (1, 2)T and >..2 = -2, (2, -l)T. The critical points at infinity satisfy the equation g2(0) = cosOQ1(cosO, sinO)- sinOP1(cosO, sinO)

= 2 cos2 0

+ 3 cos 0 sin 0 -

=

2 sin 2 9 = 0.

=

The solutions are given by 81 tan- 1(2) radians, 62 tan- 1(2) 6) = tan -I (-!) radians, and 64 = tan -I (-!) + 1f radians.

+ 1r radians,

11.2. Poincare Compactification

187

3

Figure 11.5: The function g2(8).

A plot of g2(8) is given in Figure 11.5. The ftow near a critical point at infinity is qualitatively equivalent to the ftow of the system

From Figure 11.5, the ftow is counterclockwise if tan- 1(-!) < (} < tan- 1(2). Therefore, the ftow at infinity is determined by the system

-y = -3y+2i-2,

-z = -z +2yz.

(-!,

There are critical points at A = (2, 0) and B = 0) in the yz plane. Point A is a stable node and point B is an unstable node. A phase portrait is given in Figure 11.6. Since n is odd, the antinodal points are qualitatively equivalent. A global phase portrait is shown in Figure 11.7. (b) The origin is an unstable focus and the ftow is clockwise. The critical points at infinity satisfy the equation

188

11. The Second Part of David Hilbert's Sixteenth Problem

Jl/111 1/1111 //II//

I/ II// II I//

Figure 11.6: Some trajectories in the yz plane (X = 1) that define the flow at infinity.

Figure 11.7: A global phase portrait for Example 1(a). g2(0)

= cos 0 Q 1(cos 0, sin 0) -

sin 0 P1 (cos 0, sin 0) = -(cos2 0 + sin2 0)

= 0.

There are no roots, so there are no critical points at infinity. A possible global phase portrait is given in Figure 11.8. Example 2. Show that the system given by

x• = -

2X

y- x 2 +xy + y 2 ,

has at least two Iimit cycles.

y=

x(l +x- 3y)

189

11.2. Poincare Compactification

Figure 11.8: A global phase portrait for Exarnple 1(b). There are no critical points at infinity and the flow is clockwise. Solution. TherearetwocriticalpointsatO matrix is given by

l=(-!-2x+y 1 + 2x - 3 y

= (O,O)andA = (0, l).TheJacobian -1+x+2y) - 3x ·

Now

Jo

= ( ｾ＠

ｾ Ｑ＠

)

and

JA= (

j 2 ｾ＠

) ·

Therefore, 0 is a stable focus and A is an unstable focus. On the Iine L 1 : 1 + x 3 y = 0, y = 0 and .X < 0, so the flow is transverse to L l· The critical points at infinity satisfy the equation 83(0)

= cosOQ2(cosO, sinO)- sinOP2(cosO, sinO) = cos3 0 - 2 cos 2 0 sin 0 - cos 0 sin 2 0 - sin 3 0 = 0.

A plot for 83(0) is given in Figure 11.9. There are two roots for 83(0): Ot = 0.37415 radians and 02 = 3.51574 radians. The flow near a critical point at infinity is qualitatively equivalent to the flow of the system

±y = - yz - iz + 2

2y +

y2 +

y3 -

z-

1,

2

z -2 - yz 2 - z + yz + y2 z. There is one critical point at (y, z) = (0.39265, 0), which is a col. Since n is ± z. =

even, the antinodal point is also a col, but the direction of the flow is reversed. The direction of the flow may be established by inspecting 83(0) in Figure 11.9.

11. The Second Part of David Hilbert's Sixteenth Problem

190

1.4 1.2 0.8 0.6 0.4 0.2 81 0

2

-0.2.

4

3 9

5

6

-0.4 -0.6 -0.8 -I

-1.2 -1.4

Figure 11.9: The function g3(0).

.·

ｾﾷＭＮ＠ , .

ｾ＠ (

.


with(plotsl: >plot((cos(x))A3-2*(cos(x))A2*sin(x)-cos(x)*(sin(x))A2-(sin(xllA3, >x=O .. 2*Pil; >

>#See Figure 11.7. >

>fso1ve((cos(x))A3-2*(cos(x))A2*sin(x)-cos(x)*(sin(x))A2-(sin(x))A3, >x=0 •• 1); >

># Use the same command to find a root in the interval (3,4). >

>with(DEtoolsl: >sys:=diff(x(t),t)=3*y(t)-2*(y(t))A2+2,diff(z(t),t)=z(t)-2*y(t)*z(t); >DEplot ( {sys}, [y ( tl , z (t) 1, t=-5 .. 5, [ [0, 1, 11, [ 0, 1, -11, [0, 2, 11 , [0, 2, -11, > [0, -1/2,11 , [0, -1/2, -11 , [ 0, -2, 11, [0, -2, -11, [0, 3, 11 , [0, 3, -11 , >[0,3,0.011, [0,1,0.01], [0,-2,0.011]); >

># See Figure 11.4. Note that the initial condition [0,1,01 etc. is ># not used as this causes problems when plotting the phase portrait.

11.4 Exercises 1. Draw a global phase portrait for the linear system

i = y,

i = -4x - 5y

including the ftow at infinity. 2. Draw a global phase portrait for the system

i = -3x + 4y,

-2x + 3y

y=

and give the equations defining the ftow near critical points at infinity. 3. Determine a global phase portrait for the quadratic system given by

i = x 2 + y 2 - 1,

y = 5xy- 5.

4. Draw a global phase portrait for the Lienard system •

3

x=y-x -x,

.

y=-y.

5. Draw a global phase portrait for the Lienard system i = y - x 3 + x,

y=

- y.

192

11. The Second Part of David Hilbert's Sixteenth Problem

Recommended Reading [I] C. J. Christopher and N. G Lloyd, Polynomial systems: A lower bound for the Hilbert numbers, Proc. Roy. Soc. London Ser. A, 450 (1995), 219-224. [2] H. Zoladek, Eleven smalllimit cycles in a cubic vector field, Nonlinearity, 8 ( 1995), 843-860. [3] T. R. Blows and C. Rousseau, Bifurcation at infinity in polynomial vector fields, J. Differential Equations, 104 (1993), 215-242. (4] Yu. S. Il'yashenko, Finiteness Theorems for Limit Cycles, Translations of Mathematical Monographs 94, American Mathematical Society, Providence, RI, 1991. [5] J. Ecalle, J. Martinet, J. Moussu, and J. P. Ramis, Non-accumulation des cycles-limites I, C. R. Acad. Sei. Paris Sir. I Math., 304 (1987), 375-377. [6] Li Jibin and Li Chunfu, Global bifurcation of planar disturbed Hamiltonian systems and distributions of Iimit cycles of cubic systems, Acta Math. Sinica, 28 (1985), 509-521. [7] Shi Songling, A concrete example of the existence of four Iimit cycles for plane quadratic systems, Sei. Sinica A 23 (1980), 153-158. [8] K. S. Sibirskii, The number of Iimit cycles in the neighbourhood of a critical point, Differential Equations, 1 (1965), 36-47. [9] N. Bautin, On the number of Iimit cycles which appear with the variation of the coefficients from an equilibrium point of focus or centre type, Amer. Math. Soc. Trans., 5 (1962), 396-414.

12 Limit Cycles of Lienard Systems

Aims and Objectives • To review the main results on the number of Iimit cycles of Lienard systems. • To prove two theorems conceming Iimit cycles of certain Lienard systems. On completion of this chapter the reader, should be able to • compare local and global results; • prove that certain systems have a unique Iimit cycle; • prove that a Iimit cycle has a certain shape for a large parameter value. Many autonomaus systems oftwo-dimensional differential equations can be transformed into systems of Lienard type. As discussed in Chapter 11, the results for the second part of Hilbert's sixteenth problern are rather scant. In contrast, in recent years, there have been many results published associated with Lienard systems. The major results for both global and local bifurcations of Iimit cycles for these systems are listed. A method for proving the existence, uniqueness, and hyperbolicity of a Iimit cycle is illustrated in this chapter, and the Poincare-Bendixson Theorem is applied to determine the shape of a Iimit cycle when a parameter is large.

12.1

Global Results

Consider polynomial Lienard equations of the form

12. Limit Cycles of Lienard Systems

194

x + f(x)x + g(x) =

(12.1)

0,

where f(x) is known as the damping coefficient and g(x) is called the restoring coefficient. Equation ( 12.1) corresponds to the class of systems x

(12.2)

= y, y = -g(x)- f(x)y

in the phase plane. Lienard applied the change of variable Y = y + F (x ), where f (s) ds, to obtain an equivalent system in the so-called Lienard plane: F (x) =

Jt

(12.3)

x

=Y -

F(x),

Y = -g(x).

In order for the critical point at the origin to be a nondegenerate focus or center, the conditions g(O) = 0 and g' (0) > 0 are imposed. Periodic solutions of (12.1) correspond to Iimit cycles of (12.2) and (12.3). There are many examples in the natural sciences and technology in which these and related systems are applied. The differential equation is often used to model either mechanical systems or electric circuits, and in the literature, many systems are transformed into Lienard type to aid in the investigations. For a Iist of applications to the real world, see, for example, Moreira's paper [6]. In recent years, the number of results for this class of system has been phenomenal, and the allocation of this topic to a whole chapter of the book is weil justified. These systems have proved very useful in the investigation of multiple Iimit cycles and also when proving existence, uniqueness, and hyperbolicity of a Iimit cycle. Let adenote the degree of a polynomial, and Iet H (i, j) denote the maximum number of global Iimit cycles, where i is the degree of f and j is the degree of g. The main global results for systems (12.2) and (12.3) to date are listed below: • In 1928, Lienard (see [3, Chapter 5]) proved that when ag = I and F is a continuous odd function that has a unique root at x = a and is monotone increasing for x ｾ＠ a, (12.3) has a unique Iimit cycle.

=

1 and Fis an odd polynomial of • In 1975, Rychkov [11] proved that if ag degree five, (12.3) has at most two Iimit cycles. • In 1976, Cherkas [ 10] gave conditions for a Lienard equation to have acenter. • In 1977, Lins, de Melo, and Pugh [9] proved that H(2, 1) = 1. They also conjectured that H(2m, 1) = H(2m + 1, 1) = m, where m isanatural number. • In 1988, Coppel [7] proved that H (1, 2) = 1. • In 1992, Zhang Zhifen [5] proved that a certain generalized Lienard system has a unique Iimit cycle. • In 1996, Dumortier and Chengzhi Li [4] proved that H (1 , 3) = l.

195

12.1. Global Results • In 1997, Dumortier and Chengzhi Li [3] proved that H(2, 2)

= 1.

More recently, Giacomini and Neukirch [2] introduced a new method to investigate the Iimit cycles of Lienard systems when ag = 1 and F(x) is an odd polynomial. They are able to give algebraic approximations to the Iimit cycles and obtain information on the number and bifurcation sets of the periodic solutions even when the parameters are not small. Other work has been carried out on the algebraicity of Iimit cycles but is beyond the scope of this book. Limit cycles were discussed in some detail in Chapter 5, and a method for proving the existence and uniqueness of a Iimit cycle was introduced. Another method for proving the existence, uniqueness, and hyperbolicity of a Iimit cycle is illustrated in Theorem 2. Consider the generat polynomial system x = P(x, y),

y=

Q(x, y),

where P and Q are polynomials in x and y, and define X = (P, Q) tobe the vector field. Let a Iimit cycle, say y(t) = (x(t), y(t)), have period T.

Definition 1. The quantity

Ir div(X) dt is known as the characteristic exponent.

Theorem 1. Suppose that

1. r

dtv(X)dt =

1r (aP +-aQ) 0

-

ax

ay

(x(t), y(t))dt.

Then

Ir div(X) dt < 0; y is hyperbolic repelling if Ir div(X) dt > 0.

(i) y is hyperbolic attracting if (ii)

Theorem 2. Consider the Lienard system (12.4) There exists a unique hyperbolic Iimit cycle if a 1a3 < 0.

Proof. The method is taken from the paper of Lins, de Melo, and Pugh [8]. Note that the origin is the only critical point. The flow is horizontal on the line x = 0 and vertical on the curve y = a1x + a2x 2 + a3x 3. It is not difficult to prove that a trajectory starting on the positive (or negative) y-axis will meet the negative (or positive) y-axis. The solution may be divided into three stages: I. Every Iimit cycle of system (12.4) must cross both of the lines given by

196

12. Limit Cycles of Lienard Systems

Il. System (12.4) has at least one and at most two Iimit cycles; one of them is hyperbolic.

111. System (12.4) has a unique hyperbolic Iimit cycle. Stage I. Consider the Lyapunov function given by V (x, y) = e-2a2Y (y - a2x 2 + ｾＩＮｎｯｷ＠ dV - = 2a2e-2a 2yx 2(ai + a3x 2). dt The Lyapunov function is symmetric with respect to the y-axis since V (x, y) = V(-x, y), and there is a closed Ievel curve V(x, y) = C that is tangent to both LI and L2. Since df, does not change sign inside the disk V(x, y) = C, no Iimit cycle can intersect the disk, which proves Stage I.

c

Stage TI. Suppose that there are two Iimit cycles YI as in Figure 12.1.

Y'2 surrounding the origin

X

Figure 12.1: Two Iimit cycles crossing the lines L 1 and L2. Suppose that a 1 < 0 and a3 > 0. Then the origin is unstable. Let YI be the innermost periodic orbit, which must be attracting on the inside. Therefore, { div(X) dt = { -(ai

lr

lr•

1

+ 2a2x + 3a3x 2) :::: 0.

*

Let P; and Q;, i = 0, I, 2, 3, bethepointsofintersectionofyi and n,respectively, dt = 0, and similarly for the with the lines L 1 and L2. Now 1 x dt = 1 periodic orbit Y'2· Consider the branches PoP1 and QoQI on Yl and )1'2, respectively. The ftow is never vertical on these branches. Hence one may pararnetrize the integrals by the variable x. Thus

Ir

1 (+ PoP1

-

a1

3a3x 2) d t =

Ir

1x• xo

-(ai

+ 3a3x2) dx

Yr 1(x) - F(x)

197

12.1. Global Results and

In the region xo < x < x1, the quantity -(a1 F(x) > YYt (x)- F(x) > 0. It follows that

+ 3a3x 2)

> 0 and Yn(x)-

Using similar arguments, it is not difficult to show that

Consider the branches P1 P2 and Q1 Q2 on )II and )1'2, respectively. The ftow is never horizontal on these branches. Hence one may parametrize the integrals by the variable y. Thus

and

In the region Yl < y < Y2· Xn(Y) > xy1 (y). It follows that

Using similar arguments, it is not difficult to show that

Thus adding all of the branches together,

1

div(X) dt
0 on this arc, for enough, the pointing will be as indicated in Figure 12.4.

E

small

Are 7-8. On this line, y- F(x) > 0, so i > 0. Are 8-9. On the eurve, y = F(x) with x > 0, Are 9-10. On this line, y- F(x) < 0 and

y
### Graphical iteration of the tent map ### >################ ################## ######### >

> > > >

imax:=200:mu:=2: # Initialize halfmax:=imax/2: T:=array(O .. lOOOO) :TT:=array(O .. 10000); T[0]:=0.2001:

>

> for i from 0 to imax do # Define the tent map > if T[i]>=O and T[i] T[i+ll :=mu*T[i); > elif > T[iJ>O and T[i] T[i+l] :=mu* (1-T(i]); > fi;

> od; >

> TT[O] :=[T(O] ,0] :TT[l) := (T[O] ,T[l]]:

250

14. Nonlinear Discrete Dynamical Systems

>

> for i from 1 to halfmax do # Find the coordinates > TT[2*i] :=[T[i] ,T[i]]: > TT[2*i+1]:=[T[i],T[i+1]]: > od: >

> 1 := [TT[n]$n=O .. imax]: #List the Coordinates >

> > > > > >

with(plots): M:=plot(l, x=O .. 1,y=O .. l, style=line,color=black): N:=plot(x,x=O .. 1,color=black): P:=plot(mu*x,x=O .. O.S,color=black): Q:=plot(mu*(1-x),x=0.5 .. 1,color=black): display({M,N,P,Q},labels=['x', 'T'));

>

>#See Figure 14.7(bl. >

># Program to compute Lyapunov exponents # >######################################### >

x=array(O .. 50000) :x[O) :=O.l:mu:=1:imax:=50000: for i from 0 to imax do x[i+1] :=evalf(mu*x[i) * (1-x[i))) :od: L:=O: for i from 0 to imax do L:=L+ln(abs(mu*(1-2*x[i)))): > od: > L/imax; > > > > > >

>

> ### Bifurcation diagram of the logistic map ### > ############################################### >

> > > >

with(plots): imax:=80:jmax:=350:step:=0.01: ll:=array(0 .. 10000) :pp:=array(0 .. 10000):xx :=array(O .. 10000,0 .. 10000):

>

> > > > > > > > >

for j from 0 to jmax do xx[j,Ol :=0.5: for i from 0 to imax do xx [ j, i+1] : = ( step*j l *xx [ j, i I* ( 1-xx [ j, i )) : od: ll[j] := [[(step*j),xx[j,n])$n=40 .. imax):od: LL := [seq(ll[j],j=O .. jmax)): with(plots): P1:=plot(LL,x=0 .. 4,y=-0.1 .. 1,style=point,symbol=point, tickmarks=[2,2]):

>

> > > > >

imax:=80:jmax:=200:step:=0.0025: ll:=array(O .. 10000) :pp:=array(0 .. 10000): xx:=array(O .. 10000,0 .. 10000): for j from 0 to jmax do xx[j,0):=0.5:

251

14.6. Exercises > for i from 0 to imax do > xx[j,i+l]:=(step*j+3o5)*xx[j,i]*(l-xx[j,i]) :od: >

ll[j] := [[(step*j+3o5),xx[j,n]]$n=40ooimax]:od:

> LL := [seq(ll[j],j=Oo ojmax)]:

P2:=plot(LL,x=3o4oo4,y=-Oolo ol,style=point,symbol=point, tickmarks=[2,2]): > tl:=textplot([3,-0ol, •mu'],align=BELOW):

> >

>

Plot the bifurcation diagram # display({Pl,P2,tl},color=black,labels=['', •x•],tickmarks=[2,2], > font=[TIMES,ROMAN,25],axes=FRAMED); > # See Figures 14o13, 14ol5, and 14o16o

> # >

> > > > > > > >

### The Henon map ### #####################

x:=array(Oo o10000) :y:=array(Oo o10000): a:=lo2:b:=Oo4:imax:=2000: x[OJ:=Ool:y[O]:=O:

> for i from 0 to imax do > x[i+1] :=1+y[i]-a*(x[i])A2: > y[i+1]:=b*x[i]: > od: >

> with(plots): > points:=[(x[n],y[n]J$n=300ooimax): > pointplot(points,x=-2oo2,y=-2oo2,style=point,symbol= point, > color=black); > ># See Figure 14o22(b)

14.6

Exercises

10 Consider the tent map defined by T(x) =

I

2x, 2(1 - x),

O:;:x
0. Suppose that

u = J b, or Yl = Ｍｾ＠ if y < b. The other root is then given by x2 -.Jii+lj' and Y2 -y,. This transfonnation has a two-valued inverse, and twice as many predecessors are generated on each iteration. One of these points is chosen randomly in the computer program. Recall that all unstable periodic points are on J. It is not difficult to detennine the fixed points ofperiod one for mapping (15.1). Suppose that z is a point of period one. Then

=

=

z2 - z + c = 0, which gives two solutions, either

Zl,l =

1 +.;t=4c 2

or

ZI,2

1- ｾＱＭ 4c = ----2

The stability of these fixed points can be determined in the usual way. Hence the fixed point is stable if

and it is unstable if

ld:l

< 1

258

15. Complex Iterative Maps

ld:l>l.

By selecting an unstable fixed point of period one as an initial point, it is possible to generate a Julia set using a so-called backward trainingiterative process. Julia sets define the border between bounded and unbounded orbits. Suppose that the Julia set associated with the point c = a + ib is denoted by J(a, b). As a simple example, consider the mapping (15.3)

Zn+ I =

z;.

There is one fixed point of equation (15.3) at the origin, say z*. Initial points that start wholly inside the circle of radius 1 are attracted to z*. An initial point starting on lzl = 1 will generate points that again lie on the unit circle lzl = 1. Initialpoints starting outside the unit circle will be repelled to infinity since lzl > l. Therefore, the circle lzl = I defines the Julia set J(O, 0) that is a repellor (points starting near to but not on the circle are repelled), invariant (orbits that start on the circle are mapped to other points on the unit circle), and wholly connected. The interior of the unit circle defines the basin of attraction (or domain of stability) for the fixed point atz*. In other words, any point starting inside the unit circle is attracted to z*. Suppose thatc = -0.5+0.3i in equation (15.1). Figure 15.1(a) shows apicture of the Julia set J ( -0.5, 0.3) containing 2 15 points. The Julia set J ( -0.5, 0.3) defines the border between bounded and unbounded orbits. For example, an orbit starting inside the set J(-0.5, 0.3) at zo 0 + Oi remains bounded, whereas an orbit starting outside the set J(-0.5, 0.3) atz= -I - i, for instance, is unbounded. The reader will be asked to demoostrate this in the exercises at the end of the chapter. Four of an infinite number of Julia sets are plotted in Figure 15.1. The first three are totally connected, but J {0, 1.1) is totally disconnected. In 1979, Mandelbrot devised a way of distinguishing those Julia sets that are wholly connected from those that are wholly disconnected. He used the fact that J(a, b} is connected if and only ifthe orbit generated by z--+ z2 + c is bounded. In this way, it is not difficult to generate the now famous Mandelbrot set. Assign a point on a computer screen to a coordinate position c = (a, b} in the Argand plane. The point z = 0 + Oi is then iterated under the mapping ( 15.1) to give an orbit 0 + Oi, c, c2 + c, (c2 + c} 2 + c, .... If after 50 iterations, the orbit remains bounded (within a circ1e of radius 4 in the program used here}, then the point is colored black. If the orbitleaves the circle of radius 4 after m iterations, where 1 < m < 50, then the point is co1ored black if m is even and white if m is odd. In this way, a b1ack-and-white picture of the Mandelbrot set is obtained, as in Figure 15.2. Unfortunately, Figure 15.2 does no justice to the beauty and intricacy of the Mandelbrot set. This figure is a theoretical object that can be generated to an infinite amount of detail, and the set is a kind of fractal displaying self-similarity in certain

=

259

15.2. Boundaries of Periodic Orbits

:ｾＺ＠ ; ｲＭﾷｾ｜Ｎ｣＠ Ｍｾ

0.8 0.6 0.4 0.2 lmzO -0.2

I

0.2

ｾＭ＠

...().4

}

［ＺＮｾ＠

-0.4 -0.6 ...().8 -I -1. 2 ｾＭＱＮＲ＠

-0.6 ...().8

ｾＭＰＮＸ＠

f'

. ｾ＠

...-0 .....ＴＭＮｾｯＺＬＢＧＲ｟ＰＶｳｬ＠ Rez

(a)

(b)

1.2 I .., 0.8 "" .... 0.6 0.4 0.2 lmzO -0.2 -0.4 -0.6 -0.8

'

ｾＺｹＨ＠

0.8 0.6 0.4

-0.4 -0.6 -0.8

.A.

Jt.....

ＮｌＬＭＺＧＢＷｦｑｲｾ＠ ｾ＠

\}

ｙＧｾ＠

Ｉｾ＠ ｾＮＭ＼Ｇ｜Ｑ＠

-1.6 -1.2 -0.8 -0.4

ｾＭ

-c

...1 • ..:

•.t

-I

00.2 0.6

Rez

11.2 1.6

-1.2 Ｍ｟ｬＮＺＬＢＧｾＡ｣＠ -1.4

-1

(c)

-\

-0.6 -0.20 0.20.40.60.8 I 1.21.4

Rez

(d)

Figure 15.1: Four Juliasets forthe rnapping (15.1), where J(a, b) denotes the Julia set associated with the point c = a+ib: (a) J ( -0.5, ＰＮＳＩｾ＠ (b) J(O, ｬＩｾ＠ (c) J( -1, 0), and (d) J(O, 1.1). parts and scaling behavior. One has to try to imagine a whole new universe that can be seen by zooming into the picture. For a video journey into the Mandelbrot set, the reader is once more directed to the video [4]. It has been found that this remarkable figure is a universal constant much like the Feigenbaum number introduced in Chapter 14. Some simple properties of the Mandelbrot set will be investigated in the next section.

15.2

Boundaries of Periodic Orbits

For the Mandelbrot set, the fixed points of period one may be found by solving the equation Zn+l =Zn for all n or, equivalently, fc(Z)

= z2 + c = z.

260

15. Complex Iterative Maps 1.4-

1.2 ·

0 .8 · 0.6 c 0.4 . 0.2 .

lm z

o·

--0.2 .. -0.4 -0.6 -0.8 - 1.2 . - 1.4 ｌＺｾ＠

- 2 - 1.8

- L4

- I -0.8 Re z

Figure 15.2: The Mandelbrotset (central black figure) produced using a personal computer. which is a quadratic equation of the form

z2 - z + c = 0.

(15.4) The solutions occur at zt.l

=

1 + v'f=-4c

2

and

Zl.2

=

1-

v'f=-4c 2

,

where zu is the first fixed point of period one and Zl.2 is the second fixed point of period one using the notation introduced in Chapter 14. As with other discrete systems, the stability of each period-one point is determined from the derivative of the map at the point. Now (15.5)

i6 dfc =2z=re ,

dz

where r ｾ＠ 0 and 0 :::.:: (} < 21r. Substituting from equation (15.5), equation (15.4) then becomes re;e)2 rei9 (--+c=O 2 2 .

261

15.2. Boundaries of Periodic Orbits The solution for c is (15.6) One of the fixed points, say z1,1, is stable as long as

I

dfc l dZ(ZI,I) < 1.

Therefore, using equation (15.5), the boundary of the points of period one is given by dfc

l dZ(zu)

I=

12zul

=r = 1

in this particular case. Let c = x + iy. Then from equation (15.6), the boundary is given by the following parametric equations:

x

= 21 cos(}- 41 cos(20),

1 sm . (} - 1 sm y= 2 4 . (20) .

The parametric curve is plotted in Figure 15.3 and forms a cardioid that lies at the heart of the Mandelbrot set. 0.6

0.4 0.2

lmzO

Period One

ｾＮＲ＠

- > > > > > >

Maple Commands

# Program to plot Julia sets # #####i########################

x1:=array(0 .. 1000000):y1:=array(0 .. 1000000): k:=15:iter:=2Ak: a:=-0.5:b:=0.3: die:=rand(0 .. 1): # Generates random numbers 0 or 1.

> > > > >

# Determine an unstable fixed point of period one on the set. x1[0]:=Re(0.5+sqrt(0.25-(a+I*b))): y1[0]:=Im(O.S+sqrt(0.25-(a+I*b))): 2*abs(x1[0)+I*y1[0]);

2.785907878 > > > > > > > >

# The point is unstable.

for i from 0 to iter do x:=x1[i]:y:=y1[i]: u:=sqrt((x-a)A2+(y-b)A2)/2:v:=(x-a)/2: u1:=evalf(sqrt(u+v)):v1:=evalf(sqrt(u-v)): x1[i+1]:=u1:y1[i+1J:=v1:if y1[i] m:='m': > with(plots): > pts:=[[x1[m],y1[m]J $m=O .. iter]: > pointplot(pts,style=point,symbol=point,color=black); > # See Figure 15.1. > > # Program to plot the Mandelbrot set # > ######################################

> >

mandelbrot:=proc(x,y)

> local z,m; > z:=evalf(x+I*y); > m:=O; >

to 50 while abs(zA2) < 16 do

> z:=zA2+(x+I*y);

> > > >

m:=m+1; od:

m; end:

>

> ll:=array(0 .. 100000):kk:=O:ll[0]:=[0,0];11[1]:=[0,0]: > step:=0.01:imax:=280:jmax:=140:MAX:=(imax+l)*(jmax+l): > for k from 0 to MAX do

> ll[k]:=[O,O]:od: > > for i

from 0 to imax do

15. Comple:x Iterative Maps

264 >

for j from 0 to jmax do

> x:=-2.1+step*i:y:= O+step*j: >

if mandelbrot(x,y) mod 2 = 0 then kk:=kk+2:

ll[kk] :=[x,y] :ll[kk+1] :=[x,-y]: > fi: > > od:od: > > # The nurober of points to be plotted is > lprint (kk); 51492 >

> > > > >

pts:=[ll[nn] $nn=2 .. 51492]: wi th (plots l : pointplot(pts,style =point,symbol=po int); # See Figure 15.2.

15.4 Exercises 1. Consider the Julia set given in Figure 15.1(a). Take the mapping Zn+l = ｺｾ＠ + c, where c = -0.5 + 0.3i.

(i) Iterate the initial point zo = 0 + Oi for 500 iterations and Iist the final 100. Increase the nurober of iterations. What can you deduce about the orbit? (ii) Iterate the initial point zo = -1 - i and Iist Zt to deduce about this orbit?

zw. What can you

2. Given that c = -1 + i, determine the fixed points of periods one and two for the mapping Zn+ I = ｺｾ＠ + c.

3. Considerequation (15.1). Plot theJuliasets J(O, 0), J( -0.5, 0), J( -0.7, 0), and J(-2, 0).

4. Estimate the fixed points of period one for the complex mapping

5. Determine the boundaries of points of periods one and two for the mapping 2 Zn+l = C- Zn·

Recommended Reading and Viewing [1] G W. Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, MIT Press, Cambridge, MA, 1998.

15.4. Exercises

265

[2] H.-0. Peitgen (ed.), E. M. Maletsky, H. Jürgens, T. Perciante, D. Saupe, and L. Yunker, Fraetats for the Classroom: Strategie Aetivities, Volume 2, Springer-Verlag, New York, 1992. [3] H.-0. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fraetals: New Frontiers ofSeienee, Springer-Verlag, New York, 1992. [4] H.-0. Peitgen, H. Jürgens, D. Saupe, and C. Zahlten, Fraetals: An Animated Diseussion, Spektrum Akademischer Verlag, Heidelberg, 1989; W. H. Freeman, New York, 1990. [5] H.-0. Peitgen and P. H. Richter, The Beauty of Fraetals, Springer-Verlag, Berlin, New York, Heidelberg, 1986.

16 Electromagnetic Waves and Optical Resonators

Aims and Objectives • To introduce some theory of electromagnetic waves. • To introduce optical bistability and show some related devices. • To discuss possible future applications. On completion of this chapter, the reader should be able to • understand the basic theory of Maxwell's equations; • derive the equations to model a nonlinear simple fiberring resonator; • investigate some ofthe dynamics displayed by these devices and plot chaotic attractors. As an introduction to optics, electromagnetic waves are discussed via Maxwell's equations. The reader is briefly introduced to a range of bistable optical resonators including the nonlinear Fabry-Perot interferometer, the cavity ring, the simple fiber ring, the double-coupler fiber ring, the fiber double-ring, and a nonlinear optical loop mirror (NOLM) with feedback. All of these devices can display hysteresis and all can be affected by instabilities. Possible applications are discussed in the physical world.

268

16.1

16. Electromagnetic Waves and Optical Resonators

Maxwell's Equations and Electromagnetic Waves

This section is intended to give the reader a simple general introduction to optics. Most undergraduate physics textbooks discuss Maxwell's electromagnetic equations in some detail. The aim of this section is to Iist the equations and show that Maxwell's equations can be expressedas wave equations. Maxwell was able to show conclusively that just four equations could be used to interpret and explain a great many electromagnetic phenomena. The four equations, collectively referred to as Maxwell's equations, did not originate entirely with him but with Ampere, Coulomb, Faraday, Gauss, and others. First of all, consider Faraday's law of induction, which describes how electric fields are produced from changing magnetic fields. This equation can be written as

rcJ E.dr =- öf/1, Öt where E is the electric field strength, r is a spatial vector, and f/1 is the electric potential. This equation may be written as

rcJ E.dr = _!_öt lls{{

B.dS,

where B is a magnetic field vector. Applying Stokes's theorem,

ffs VA

E.dS = -

:t ffs

B.dS.

Therefore,

öß VAE=-öt'

(16.1)

which is the pointform of Faraday's law of induction. Ampere 's law describes the production of magnetic fields by electric currents. Now

fc

H.dr

= ffs J.dS,

where His a magnetic field vector and J is the current density. By Stokes's theorem,

fc

H.dr =

ffs VA

H.dS =

ffs

J.dS.

Therefore,

VAH=J. Maxwell modified this equation by adding the time rate of change of the electric flux density (electric displacement) to obtain

(16.2)

öD

VAH=J+-, Öl

16.1. Maxwell's Equations and Electromagnetic Waves

269

where D is the electric displacement vector. Gauss 's law for electricity describes the electric field for electric charges, and Gauss's law for magnetism shows that magnetic field lines are continuous without end. The equations are p (16.3) V.E=-, €0

where p is the charge density and €o is the permittivity of free space (a vacuum), and (16.4)

V.B=O.

In using Maxwell's equations (16.1)-(16.4) above and solving problems in electromagnetism, the three so-called constitutive relationsarealso used. Theseare B = J.LH = J.LrJ.LoH,

D = €E = €r€oE,

and

J = aE,

where J.Lr and J.Lo are the relative permeabilities of a material and free space, respectively, €r and €Q are the relative permittivities of a material and free space, respectively, and a is conductivity. If E and H are sinusoidally varying functions of time, then in a region of free space, Maxwell 's equations become V.E = 0,

V.H = 0,

VA E

+ iwJ.LoH =

0,

and

VA H- icu€oE = 0.

The wave equations are obtained by taking the curls of the last two equations. Thus V2 E

+ €OJ.Low 2 E =

0

and

V2 H

+ €OJ.Low 2 H =

0,

where w is the angular frequency of the wave. These differential equations model an unattenuated wave traveling with velocity 1 c=--,

.,ßöiiö

where c is the speed of light in a vacuum. The field equation E(r, t) = Eoexp[i(wt- kr)]

2r

satisfies the wave equation, where k = is the wave vector and A. is the wavelength of the wave. The remarkable conclusions drawn by Maxwell were that light is an electromagnetic wave and that its properlies can all be deduced from his equations. The electric fields propagating through an optical fiber loop will be investigated in this chapter and Chapter 17, where we investigate optical bistability. Similar equations are used to model the propagation of light waves through different media including a dielectric (a nonconducting material whose properlies are isotropic); see Section 16.2. In applications to nonlinear optics, the MaxwellDebye or Maxwell-Bloch equations are usually used, but that theory is beyond the scope of this book. Interested readers are referred to [3], [8], and the research papers listed at the end of this chapter, for example.

270

16.2

16. Electromagnetic Waves and Optical Resonators

Historical Background of Optical Resonators

In recent years, there has been a great deal of interest in optical bistability due to its potential applications in high-speed all-optical signal processing and alloptical computing. Indeed, in 1984, Smith [9] published an article in Nature with the enthralling title "Towards the Optical Computer," andin 1999, Matthews [1] reported on work carried out by A. Wixforth and his group on the possibility of optical memories. Bistahle devices can be used as logic gates, memory devices, switches, and differential amplifiers. The electronic components used nowadays can interfere with one another, need wires to guide the electronic signals, and carry information relatively slowly. Using light beams, it is possible to connect all-optical components. There is no interference, Jenses and mirrors can be used to communicate thousands of channels of information in parallel, the informationcarrying capacity-the bandwidth-is enormous, and there is nothing faster than the speed of light in the known universe. In 1969, Szöke et al. [15] proposed the principle of optical bistability and suggested that optical devices could be superior to their electronic counterparts. As reported in Chapter 7, the two essential ingredients for bistability are nonlinearity and feedback. For optical hysteresis, nonlinearity is provided by the medium as a refractive (or dispersive) nonlinearity or as an absorptive nonlinearity or as both. Refractive nonlinearities alone will be considered in this chapter. The feedback is introduced through mirrors or fiber loops or by the use of an electronic circuit. The bistable optical effect was first observed in sodium vapor in 1976 at Bell Laboratories, and a theoretical explanationwas provided by Felber and Marburger [14] in the same year. Nonlinearity was due to the Kerreffect (see Section 16.3), which modulated the refractive index of the medium. Early experimental apparatus for producing optical bistability consisted of hybrid devices that contained both electronic and optical components. Materials used included indium antimonide (lnSb), gallium arsenide (GaAs), and tellurium (Te). By 1979, micron-sized optical resonators had been constructed. A fundamental model of the nonlinear Fabry-Perot interferometer is shown in Figure 16.1. An excellent introduction to nonlinearity in optics is provided by the textbook of Agrawal [4]. A block diagram of the first electrooptic device is shown in Figure 16.2 and was constructed by Smith and Turner in 1977 [13]. Nonlinearity is induced by the

I

Nonlinear medium

T

R Figure 16.1: A Fabry-Perot resonator: I, R, and T stand for incident, reflected, and transmitted intensities, respectively.

16.2. Historical Background of Optical Resonators

ｷｾｴｯ＠ I

=

Amplifier

[

R

271

Fabry:Perot Resonator

T Beamsplitter

Figure 16.2: The first electrooptic device to display bistability. Fabry-Perot interferometer and a He-Ne (helium-neon) laser is used at 6328A. A bistable region is observed for a small range of parameter values. In theoretical studies, Ikeda, Daido, and Akimoto [12] showed that optical circuits exhibiting bistable behavior can also contain temporal instabilities under certain conditions. The cavity ring (CR) resonator, first investigated by Ikeda, consists of a ring cavity comprising four mirrors that provide the feedback and containing a nonlinear dielectric material; see Figure 16.3. Light circulates around the cavity in one direction and the medium induces a nonlinear phase shift dependent on the intensity of the light. Mirrors M 1 and M2 are partially reflective, while mirrors M3 and M4 are 100% reflective.

R I

Figure 16.3: The CR resonator containing a nonlinear dielectric medium. Possible bifurcation diagrams for this device are shown in Figure 16.4. In Figure 16.4(a), the bistable region is isolated from any instabilities, but in Figure 16.4(b), instabilities have encroached on the bistable cycle. The length L is different in the two cases and hence so is the cavity round-trip time (the time it takeslight to complete one loop in the cavity). In recent years, there has been intense research activity in the field of fiber optics. Many A-level physics textbooks now provide an excellent introduction to the subject, and [4] provides an introduction to nonlinear optics. The interest in this chapter, however, lies solely in the application to all-optical bistability. A block diagram of the simple .fiber ring (SFR) resonator is shown in Figure 16.5. lt has recently been shown that the dynamics of this device are the same as those for the CR resonator (over a limited range of initial time) apart from a scaling. The

16. Electromagnetic Waves and Optical Resonators

272

8 Output

S Output

:L:=::.

6

6

0

I

3

2

4

Ｚｾ＠

S

0

I

3

2

4

S

(b)

(a)

Figure 16.4: Possible bifurcation diagrams for the CR resonator: (a) An isolated bistable region. (b) Instabilities within the bistable region; S represents stable behavior, P is period-undoubling, and C stands for chaos.

Fiber coupler

...

Eout

Figure 16.5: A schematic of the SFR resonator. The input electric field is Ein and the output electric field is Eout· first all-optical experiment was carried out using a single-mode fiber in a simple loop arrangement, the fiber acting as the nonlinear medium [10]. In mathematical models, the input electric field is given as Ein(t)

= ｾｪＨｴＩ･ｩｷＬ＠

where ;j represents a complex amplitude (which may contain phase information) and w is the circular frequency of the light. In experimental setups, for example, the light source could be a Q-switched YAG Iaser operating at 1.06JLm. The optical fiber is made of fused silica and is assumed to be lossless. An analysis of the SFR resonator will be discussed in more detail in Section 16.3, and the stability ofthe device wiii be investigated in Chapter 17. The double-coup/er fiberring resonatorwas investigated by Li and Ogusu [2] in 1998; see Figure 16.6. It was found that there was a similarity between the dynamics displayed by this device and the Fabry-Perot resonator in terms of trans-

16.3. The Nonlinear Simple Fiber Ring Resonator

E

273

Fiber coupler

Figure 16.6: The double-coupler fiberring resonator: Ein is the input field amplitude, is the reflected output, and E!ut is the transmitted output.

E!t

mission and reflection bistability. It is possible to generate both clockwise and counterclockwise hysteresis loops using this device. An example of a counterclockwise bistable cycle is given in Figure 16.4(a). The reader will be asked to carry out some mathematical analysis for this device in the exercises at the end of the chapter (Section 16.6). In 1994, Ja (5] presented a theoretical study of an optical fiber double-ring resonator, as shown in Figure 16.7. Ja predicted multiple bistability of the outpul intensity using the Kerr effect. However, instabilities were not discussed. It was proposed that this type of device could be used in new computer logic systems where more than two logic states are required. In principle, it is possible to link a number of loops of fiber, but instabilities are expected to cause some problems.

Figure 16.7: A fiberdouble-ring resonator with two couplers. The nonlinear opticalloop mirror (NOLM) with feedback [6] has been one of the most novel devices for demonstrating a wide range of all-optical processing functions including optical logic. The device is shown in Figure 16.8. Note that the beams oflight are counter-propagating in the )arge loop but not in the feedback section and that there are three couplers. All of the devices discussed thus far can display bistability and instability, leading to chaos. To aid in understanding some of these dynamics, the SFR resonator will now be discussed in some detail.

16.3

The Nonlinear Simple Fiber Ring Resonator

Consider the all-optical fiber resonator as depicted in Figure 16.9 and define the slowly varying complex electric fields as indicated.

274

16. Electromagnetic Waves and Optical Resonators

Figure 16.8: A schematic of a NOLM with feedback.

...

Ein

K:I-JC

ü '

Eout

...

L

Figure 16.9: The SFR resonator: The e1ectric fie1d entering the fiberring is labeled E1 and the electric field leaving the fiberring is labeled E2. The coupler splits the power intensity in the ratio K : 1 - K. Note that the power P and intensity I are related to the electric field in the following way: If the electric field crosses the coupler, then a phase shift is induced, which is represented by a multiplication by i in the equations. Assurne that there is no loss at the coupler. Then across the coupler, the complex field amplitudes satisfy the following equations:

(16.5) and (16.6) where " is the power splitting ratio at the coup1er. Consider the propagation from E1 to E2. Then (16.7)

16.3. The Nonlinear Simple Fiber Ring Resonator

275

where the totalloss in the fiber is negligible (typically about 0.2dB/km) and

The linear phase shift is given by

4>L. and the nonlinear phase shift due to propagation 2H r2L

is

2

4>NL = --IEII , A.oAerr

where A.o is the wavelength of propagating light in a vacuum, Aeff is the effective core area of the fiber, L is the Jength of the fiber loop, and r2 is the nonlinear refractive index coefficient of the fiber. lt is weH known that when the optical intensity is large enough, the constant r2 satisfies the equation r = ro +r2/ = ro

'2'0 2 + -lEI =

Ｒｾｯ＠

p ro+r2-. Aerr

where r is the refractive index of the fiber, ro is the linear value, I is the instantaneous optical intensity, P is the power, and Aeff is the effective cross-sectional area of the fiber. If the nonlinearity of the fiber is represented by this equation, then the fiber is said to be of Kerr type. In most applications, it is assumed that the response time of the Kerr effect is much less than the time taken for light to circulate once in the loop. Substitute (16.7) into equations (16.5) and (16.6). Simplify to obtain E1 (t)

= i.Jl=ICEin(t) +,.fiEl (t- tR)eit/l(t-tR),

'f'

where tR = is the time taken forthe lighttocompleteone Ioop, r is the refractive index, and c is the velocity oflight in a vacuum. Note that this is an iterative formula for the electric field amplitude inside the ring. This expression can be written more conveniently as an iterative equation of the form (16.8)

En+I

2Hr2L 2 = A +BEn exp ( i ( --IEnl + 4>L ) ) , >..oAerr

where A = i .Jf="KEin• B = .fi, and E J is the electric field amplitude at the jth circulation around the fiber loop. Typical fiber parameters chosen for this system are >..o = 1.55 x 10-6 m, r2 = 3.2 x w- 20 m2 w- 1, Aerf = 30JLm 2 , and L = 80m. Equation (16.8) may be scaled without loss of generality to the simplified equation (16.9) Some of the dynamics of equation (16.9) will be discussed in the next section.

16. Electromagnetic Waves and Optical Resonators

276

16.4

Chaotic Attractors and Bistability

Split equation (16.9) into its real and imaginary parts by Setting En = Xn and set (/JL = 0. The equivalent real two-dimensional system is given by

Xn+l

= A + B(xn cos IEnl 2 -

+ iyn.

Yn sin IEnl 2)

(16.10) where IBI < 1. This system is one version of the so-called lkeda map. As with the Henon map introduced in Chapter 14, the Ikeda map can have fixed points of all periods. In this particular case, system ( 16. 10) can have many fixed points of period one depending on the parameter values A and B. Example 1. Determine and classify the fixed points of period one for system (16.10) when B = 0.15 and (i) A = 1;

(ii) A = 2.2. Solution. The fixed points of period one satisfy the simultaneaus equations

x = A

+ Bx cos(x 2 + i> -

By sin(x 2

+ y2)

and (i) When A = 1 and B = 0.15, there is one solution at xu ｾ＠ 1.048, Yl,l ｾ＠ 0.151. The solution is given graphically in Figure 16. 10. To classify the critical point P* = (x 1,1, Yl.l ), consider the Jacobian matrix i)p

J(p*) =

(

ｾ＠

i)x

ｾ＠

i)p ay i)y

) p•

The eigenvalues of the Jacobian matrix at P* are A.t ｾ＠ -0.086 + 0.123i and >..2 ｾ＠ -0.086- 0.123i. Therefore, P* is a stable fixed point ofperiod one. 0.15, there are three points of period one, as 2.2 and B {ii) When A the graphs in Figure 16.11 indicate. The fixed points occur approximately at the (1.968, -0.185). (2.134, -0.317), and L (2.562, 0.131), M points U ± 0.039i; -0.145 = A.1.2 are U for eigenvalues the matrix, Using the Jacobian for L are eigenvalues the and the eigenva1ues forM are A.t = 1.360, >..2 = 0.017; A.t = 0.555, A.2 = 0.041. Therefore, U and L are stable fixed points of period one, while M is an unstable fixed point of period one. These three points are located within a bistable region of the bifurcation diagram given in Chapter 17. The point U lies on the upper branch of the hysteresis loop and the point L lies on the lower branch. Since M is unstable it does not appear in the bifurcation diagram but is located between U and L.

=

=

=

=

=

16.4. Chaotic Attractors and Bistability

277

-2

Figure 16.10: The fixed points ofperiod one are determined by the intersections of the two curves, x 1 + 0.15x cos(x 2 + y 2) - 0.15y sin(x 2 + y 2) and y 2 2 0.15x sin(x + y ) + 0.15y cos(x 2 + y 2).

=

=

4

2

YO

-2

-4

0

Figure 16.11 : The fixed points of period one are determined by the intersections of the two curves, x = 2.2 + 0.15x cos(x 2 + y 2) - 0.15y sin(x2 + y2) and y = 0.15x sin(x 2 + y 2) + 0.15y cos(x 2 + y 2).

16. Electromagnetic Waves and Optical Resonators

278

Example 2. Plot iterative maps for system (16.10) when B = 0.15 and (a) A = 5; (b) A = 10.

Solution. See Figure 16.12. Problem. Use the Maple program given at the end of the chapter (Section 16.5) to show that when A = 1, there is a unique fixed point of period one; when A = 2.2, there are two stable fixed points of period one; and when A = 2.4, there is a stable fixed point of period two. In the cases where A = 5 and A = 10, there are chaotic attractors. All of this information can be summarized on a bifurcation diagram that will be achieved in Chapter 17. Theorem 1. The circle ofradius ｾＭＺ＠

centered at Ais invariantfor system (16.10).

Proof. Suppose that a generat initial point in the Argand diagram is taken to be En. Then the first iterate is given by

En+l = A + BEneiiE"I 2 •

The second iterate can be written as En+2 = A + BEn+JeiiE"+li 2 = A + B ( A + BEneiiE"1 2 ) i1E"+J12.

Thus

En+2 = A + ABeiiEn+liz + B2Enei) ei1En+212.

Therefore, En+J = A + ABeiiEn+21 2+ AB2ei(1En+J1 2+1En+21 2) + B3 Enei(IEni 2+1En+JI 2+1En+21 2).

A generat expression for the Nth iterate En+N is not difficult to formulate. Hence En+N = A

+ ABeiiEn+N-112 + AB2ei(IEn+N-212+1En+N-112) + ... + ABN-l exp

(i ｾ＠

IEn+i1 2) J=l

+ BN Enexp

(i ｾ＠

1En+i1 2). ]=0

AsN--+ oo, BN--+ 0 since 0 < B < 1. Set Rj = IEn+N-jl 2. Then IEn+N- Al= IABeiRJ

+ AB2ei + ... + ABN-Ieij_

279

16.5. Maple Commands Since

lzt + Z2 + · ·· + Zml ｾ＠ lztl + lz2l + ·· · + lzml and lei 9 1 = 1, !En+N - Al ｾ＠

!ABI+ IAB 2 1+ · · · + IABN-ll.

This fonns an infinite geometric series as N -+ oo. Therefore, !En+N- ａｬｾ＠

IABI

1_

s·

Therefore, the disk given by IE - Al = ､ｾＭ Ｘ Ｘ ＾＠ is positively invariant for system (16.10). Theinvariant disks in two cases are easily identified in Figures 16.12(a) and (b).

8.5

9

9.5

10

10.5

II

11.5

Jt

(a)

Figure 16.12: Thechaoticattractor swhen(a)A (5000 iterates).

16.5

(b)

= 5(5000iterates)and(b)A = lO

Maple Commands

> ## Iterative plots for the scaled SFR resonator ## > > El:=array(0 .. 10000) :xl:=array(O .. 10000):yl:=array(0 .. 10000): > maxrn:=SOOO:B:=O. lS:A:=lO:El[OI :=A:xl[OI:=A:yl[OI :=0:

> >

for i from 0 to maxrn do

> El [ i+ll : =evalf (A+B*El [i 1*exp (I* (abs (El (i 1)) A2 l l :

> xl[i+li:=Re(El[i+ lll :yl[i+ll :=Im(El[i+lll: >

> > > > > >

>

od:

## Plot the points ## wi th (plots) : pointsl:=[(xl[nl,y l[nll$n=180 .. maxrnl: pointplot(pointsl, style=point,symbo l=point); # See Figure 16 . 12(b).

280 > ## > ##

16. Electromagnetic Waves and Optical Resonators Imp1icit p1ots to find points of period one ## for the scaled SFR resonator ##

> > with(p1ots):B:=0.15:A:=2.2: > fso1ve({A+B*(x*cos((xA2+yA2))-y*sin((xA2+yA2)))-x,B*(x*sin((xA2

> +yA2))+y*cos((xA2+yA2)))-y},{x,y},{x=2.4 .. 2.7,y=-O.S .. 0.5});

{X = 2.561744678, y = .1310896431} > 11:=irnp1icitplot(A+B*(x*cos((xA2+yA2))-y*sin((xA2+yA2)))-x,x=0 .. 4, > y=-5 .. 5,grid=[100,100],color=b1ack):

> 12:=imp1icitplot(B*(x*sin((xA2+yA2))+y*cos((xA2+yA2)))-y,x=0 .. 4, > y=-5 .. 5,grid=[100,100],color=b1ack): > display({ll,l2}); > # See Figure 16.11.

16.6 Exercises l. Determine the number of fixed points of period one for system (16.10) when B 0.4 and A 3.9 by plotting the graphs of the simultaneous equations.

=

=

2. Plot iterative maps for equation (16.8) using the parameter values given in

the text when K = 0.0225 and (i) Ein

= 4.5;

(ii) Ein = 6.3; (iii) Ein = 11.

3. Given that prove that the inverse map is given by En+l =

4. Given that

(EB A) exp (-iiE 11 -

11 -

ß2

Al 2 )

.

En+l = A + BE"ei1Enl2'

sbow that the steady-state solution E11 +J = E11 = Es satisfies the equation 1 ( 1 + B 2 - IEsl A2 ) cosiEsl 2 = 28 2

•

16.6. Exercises

281 lC: 1-K

Ln

Figure 16.13: Schematic of a double-coupler fiberring resonator. 5. Consider the double-coupler nonlinear fiberring resonator shown in Figure 16.13. Suppose that ER(t) = .../KEin(t)

+ i.Jl=KE4(t),

E1(t) = i.Jl=KEin(t)

+ ..,fKE4(t),

E2(t) = E1 (t- tR)eiiPt(t-tR),

1rr2L J..Aerr

2

tPI(t- IR)= --IEJ(t- tR)i , E3(t) = .../KE2(t), Er(t) = i../f=KE2(t), E4(t) = E3(t - tR)eiiPl(t-tR),

1rr2L J..Aerr

2

L)],

(17.1)

where E" is the slowly varying field amplitude, A = i .Jl=KEin is related to the input, B = ,JK, where K is the power coupling ratio, and if>L is the linear phase shift suffered by the electric field as it propagates through the fiber loop. To simplify the linear stability analysis, there is assumed to be no loss at the coupler and the phase shift if>L is set to zero. The effect of introducing a linear phase shift will be discussed later in this chapter. Suppose that Es is a stable solution of the iterative equation (17.1). Then

Es= A Therefore,

+ BEsei1Es12.

A =Es[ I - B(cos(1Es1 2) + i sin 1Es1 2)].

Using the relation lzl 2 = zz*, where z* is the conjugate of z, IA1 2 = (Es[I-B(cos(1Esl 2 )+i sin 1Esi 2 )])(E5[l-B(cos(1Es 1 2)-i sin IEs1 2 )]). Hence

(17.2) The stationary solutions of system (17.1) are given as a multivalued function of A satisfying equation (17.2). This gives a bistable relationship equivalent to the graphical method, which is weil documented in the literature; see, for example, [4], [5], and [6]. Differentiale equation (17 .2) to obtain

(17.3) To establish where the stable solutions become unstable, consider a slight perturbation from the stable situation in the fiber ring, and Iet

(17.4)

ｅＢＨｴＩ］ｳＫｾ

Ｑ ＨｴＩ＠

and

ｅ Ｑ ＫｊＨｴＩ］ｅｳｾｮＬ＠

285

17.1. Linear Stabllity Analysis

is a small time-dependent perturbation to Es. Substitute (17.4) into where ｾｮＨｴＩ＠ (17.1) to get

= A + B(Es + ｾｮＩ＠

Es+ ｾｮＫｬ＠

exp[i(Es + gn)(Es + g;)],

so (17.5)

Es+ ｾｮＫｬ＠

= A + B(Es + ｾｮＩ＠

exp[i 1Esl2l exp[i ＨｅｳｾＺ＠

+ ｾｮ＠ Es+ lgn 12)].

Take a Taylor series expansion of the exponential function to obtain

･ｸｰ｛ｩＨｅｳｾＺ＠

+ ｾｮｅｳ＠

+ ｬｾｮ Ｒ

Ｉ｝＠ = 1 + ｩＨｅｳｾＺ＠

+ ｾｮｅｳ＠

+ lgnl 2 )

+ ｩ Ｒ ＨｅｳｧＺＫｾ［ｬｮ

+ ....

Equation ( 17 .5) then becomes

lgnore the nonlinear terms in ｾｮＮ＠

Es+ ｾｮＫｬ＠

Ｒ Ｉ Ｒ＠

= A + B(Es + ｾｮＩ･ｸｰ｛ｩＱｅｳ

｝ＨＱ＠ Ｒ

+ ｩｅｳｾＺ＠

+ ｾｮｅｳ＾ﾷ＠

Since A =Es- BEs exp[iiEs1 2 J, the equation simplifies to

ｾｮＫｬ＠

(17.6)

Ｒ

= B(gn +i1Esl 2gn ＫｩＨｅｳＩ

Ｒ

ｾ［Ｉ･ｸｰＨｩＱｅｳ

ＩＮ＠

Since g is real, it may be split into its positive and negative frequency parts as follows:

(17 .7)

gn = E+eA.t + E_eA. "t

and

= E+eA.(t+tR) + E_eA. *(t+tR), ｾｮＫｬ＠

where IE+I and IE-1 are much smaller than IEsl. tR is the fiberring round-trip time, and J.. is the amplification rate of a small fluctuation added to a stable solution. Substitute equation (17.7) into (17.6). Then the validity of (17.6) at all timest requires that

E+eA.tR = B(E+ + i1Esi 2 E+ + ｩｅｪｾＩ･ｸｰＨｳｬ ｅｾ･ａＮｴｒ＠

= ｂＨｅｾ＠

ＭｩＱｅｳ Ｒ

ｅｾ＠

Ｒ

ＩＬ＠

-i(Es) 2E+)exp(-i1Es1 2)

or, equivalently, 2 ( ,8(1 + i1Es1 ) - eA.tR -iß*(Es)2

i,BEj

) ( E+ ) ( 0 ) ,8*(1- i1Es12)- eA.tR ｅｾ＠ = 0 •

where ,8 = Bexp(i1Esl 2). To obtain a valid solution, the characteristic equation must be solved:

286

17. Analysis of Nonlinear Optical Resonators

Substituting from equation ( 17 .3), the characteristic equation becomes (17.8)

e2'AtR- e'AIR

(t +

B2-

diAI2) + B2 = 0 · diEsl 2

Let D = ｾＧ［ Ｒ ＠ The stability edges for Es occur where e'A1R = + 1 and e'A1R = -1 since this is a discrete mapping. Using equation (17 .8). This yields the conditions D+I

=0

and

D-1 = 2(1

+ B 2).

Thus the system is stable as long as

0 < D < 2(1

(17.9)

+ B 2 ).

The condition D = 0 marks the boundary between the branches of positive and negative slope on the graph of 1Esi 2 versus IAI 2and hence defines the regions where the system is bistable. Thus the results from the graphical method match with the results from the linear stability analysis. The system becomes unstable at D-1· the boundary where D lt is now possible to apply four different methods of analysis to determine the stability of the electric field amplitude in the SFR resonator. Linear stability analysis may be used to determine both the unstable and bistable regions. The graphical method is redundant in this case, and the two iterative methods introduced in Chapter 9 can be used to plot bifurcation diagrams. Since the results from the graphical method are included in the results from the linear stability analysis, only three of the four methods will be compared in the next section.

=

17.2

Instabilities and Bistability

In the previous section, the results from the linear stability analysis established that system (17.1) is stable as long as equation (17.9) is satisfied. A possible stability diagram for system (17.1) is given in Figure 17.1, which shows the graph of D = ｾＧＺ Ｒ＠ and the bounding Iines D+I = 0 and D_, = 2(1 + B 2 ) when

B

= 0.15.

Table 17 .I lists the first two bistable and unstable intensity regions for the SFR resonator (in Watts per meter squared) for a range of fixed values of the parameter B. The dynamic behavior of system (17.1) may also be investigated by plotting bifurcation diagrams, which may be plotted using either the first or second iterative methods defined in Chapter 9. To observe any hysteresis, one must, of course, use the second iterative method, which involves a feedback. The method developed by Bischofherger and Shen in 1979 [5] foranonlinear Fabry-Perot interferometer is modified and used here for the SFR resonator. The input intensity is increased to a maximum and then decreased back to zero, as depicted in Figure 17 .2. In this case, the simulation consists of a triangular pulse entering the ring configuration, but it is not difficult to modify the Maple program to investigate Gaussian input pulses.

287

17.2. lnstabilities and Bistability 6

D 4

2

4

6

8 10 12 14 16 18 20 lEj

Figure 17.1: Stability diagram for the SFR resonator when B 0.0225). The systern is stable as long as 0 < D < 2(1 + B 2).

= 0.15

(K

=

Table 17.1: The first two regions of bistability and instability cornputed for the SFR resonator to three decirnal places using a linear stability analysis. B

First bistable region

First unstable region

Second bistable region

Second unstable region

0.05 0.15 0.3 0.6 0.9

10.970-11.038 4.389-4.915 3.046-5.951 1.004-8.798 0.063-12.348

12.683-16.272 5.436-12.007 1.987-4.704 1.523-7.930 I. 759-11.335

16.785-17.704 9.009-12.765 6.142-16.175 2.010-24.412 0.126-34.401

17.878-23.561 9.554-20.510 3.633-15.758 1.461-24.090 0.603-34.021

A 2 ;wm-2

A 2 ;wm-2

A 2 ;wm-2

A 2 ;wm-2

The input intensity is increased linearly up to 16Wrn-2 and then decreased back down to zero. Figure 17.2 shows the output intensity and input intensity against the nurnber of passes around the ring, which in this particular case was 4000. To observe the bistable region, it is necessary to display the rarnp-up and rarnp-down parts of the diagram on the sarne graph, as in Figure 17 .3(b). Figure 17.3 shows a gallery of bifurcation diagrarns corresponding to the pararneter values used in Table 17.1 produced using the second iterative rnethod. The diagrarns rnake interesting comparisons with the results displayed in Table 17 .1. A nurnerical investigation has revealed that for a small range of values close to B = 0.15 (see Figure 17 .3(b)), the SFR resonator could be used as a bistable device. Unfortunately, for most values of B, instabilities overlap with the first bistable region. For exarnple, when B = 0.3 (Figure 17.3(c)), the first unstable region between 1.987Wm-2 and 4.704Wm- 2 intersects with the first bistable region between 3.046Wm- 2 and 5.951Wrn- 2 • Clearly, the instabilities have affected the

17. Analysis of Nonlinear Optical Resonators

288 20 Output

15

10

5

Number of passes

Figure 17.2: Bifurcation diagram when B = 0.15. Plot of triangular input and output intensities against nurober of ring passes for the SFR resonator. bistable operation. In fact, the hysteresis cycle has failed to materialize. Recall that B = ,JK, where " is the power coupling ratio. As the parameter B gets larger, more of the input power is circulated in the ring, and this causes the system to become chaotic for low input intensities. The bifurcation diagrams are disp1ayed in Figures 17.3(d) and (e). These cases will not be discussed here since there are no isolated bistable regions. The first iterative method can be employed to show regions of instability. Note, however, that bistable regions will not be displayed since there is no feedback in this method. It is sometimes possible for a small unstable region to be missed using the second iterative method; the steady state remains on the unstable branch until it becomes stable again. Thus in a few cases, the first iterative method gives results that may be missed using the second iterative method. As a particular example, consider system (17 .1) where B = 0.225. Results from a linear stability analysis indicate that there should be an unstable region in the range 2.741-3.416Wm-2 . Figure 17.4(a) shows that this region is missed using the second iterative method, whereas the firstiterative method (Figure 17.4(b)), clearly displays period-two behavior. In physical applications, one would expect relatively small unstable regions to be skipped, as in the former case. Consider the complex iterative equation (17.10)

--IEnl 2 - l/JL )] ' En+l = ivｾ＠ 1 - KEin+ ./KEn exp [ j (2rrn2L .A.oAeff

which was derived in Chapter 16. Equation (17 .10) is the iterative equation that models the electric field in the SFR resonator. Typical fiber parameters chosen for

289

17.2. Instabilities and Bistability 28 26 2 # The tau curve # > ################# > > # Evaluate the tau function using equation (19.3). >

> > > > >

wi th (plots) : p1:=1/3:p2:=2/3: tau:=(ln(p1·x+p2·x))/(ln(3)); plot(tau,x=-10 .. 10); # See Figure 19.2(a)

> >

> # The Dq spectrum # > ################### >

> # Use equation (3) to find Dq from tau. > > Dq1:=(tau/(l-x)): > Dl:=p1ot(Dq1,x=-20 .. 0.9999l: > D2:=plot(Dql,x=1.0001 .. 20l: > display({Dl,D2}); >#See Figure 19.2(b). >

> > # The f-alpha spectrum # > ######################## >

> # Find alpha and f in terms of s (t here) and then use > # a parametric plot. > > with(plotsl: > k:=SOO:pl:=l/3:p2:=2/3: > plot([(t*ln(pl)+(k-t)*ln(p2))/(k*ln(l/3)), > -(ln(binomial(k,t)))/(k*ln(l/3)), > t=O .. SOO]); > # See Figure 19.2(c)

19.4 Exercises I. Prove that

L:f=l

. Pi ln(p;) D I = Ilffi ==---'-l-+0 -ln(l)

by applying L'Höpital's rule to equation (19.3). 2. Plot r(q) curves and the Dq and f(a) spectra for the multifractal Cantor set described in Example I when

327

19.4. Exercises (i) PI

= ｾ＠

(ii) PI = (iii) PI

and P2

= ｾ［＠

! and P2 =

= ｾ＠

and P2

};

= J.

3. A multifractal Koch curve is constructed as in Chapter 18, but now a mass is distributed, as depicted in Figure 19.9. Plot the f(a) spectrum when PI = j and P2 =

!·

Figure 19.9: The motif used to construct the Koch curve multifractal, where 2pl +2P2 = 1. 4. A multifractal square Koch curve is constructed as in Chapter 18, but now a mass is distributed, as depicted in Figure 19.10. Plot the r(q) curve and the Dq and f(a) spectra when PI = ｾ＠ and P2 = j.

Figure 19.10: The motif used to construct the Koch curve multifractal, where 3pl +2P2 = 1.

5. A multifractal Koch curve is constructed as in exercise 3, but now a mass is distributed, as depicted in Figure 19.11. Determine a1 and fs·

Figure 19.11: The motif used to construct the Koch curve multifractal, where PI + P2 + P3 + P4 = 1.

19. Multifractals

328

Recommended Reading [1] M. Alberand J. Peinke, lmproved multifractal box-counting algorithm, virtual phase transitions, and negative dimensions, Phys. Rev. E, 57-5 (1998), 5489-5493.

[2] K. J. Falconer and B. Lammering, Fractal properlies of generalized Sierpirtski triangles, Fractals, 6-1 (1998), 31-41. [3] K. J. Falconer, Techniques in Fraetat Geometry, John Wiley, New York, 1997.

[4] S. L. Mills, C. Liauw, G Lees, and S. Lynch, Assessment of filler dispersion using image analysis methods, in the extended abstracts of the 1997 MOFFISIFILPLAS International Conference on Filled Polymersand Fillers (Euro-Fillers '97), Manchester, UK, 1997,259-262. [5] Li Hua, D. Ze-jun, and Wu Ziqin, Multifraetat analysis of the spatial distribution of secondary-electron emission sites, Phys. Rev. B, 53-24 (1996), 16631-16636.

[6] V. Silberschmidt, Fraetat and multifractal characteristics of propagating cracks, J. Physique IV, 6 (1996), 287-294. [7] J. Mach, F. Mas, and F. Sagues, Two representations in multifractal analysis, J. Phys. A, 28 (1995), 5607-5622. [8] J. Muller, 0. K. Huseby, and A. Saucier, lnftuence of multifractal scaling of pore geometry on permeabilities of sedimentary rocks, Chaos, Solitons, Fractals, 5-8 (1995), 1485-1492. [9] N. Sarkar and B. B. Chaudhuri, Multifraetat and generalized dimensions of gray-tone digital images, Signal Process., 42 (1995), 181-190. [10] S. Blacher, F. Brouers, R. Fayt, and P. Teyssie, Multifraetat analysis: A new method for the characterization of the morphology of multicomponent polymer systems, J. Polymer Sei. B, 31 (1993), 655-662. [ 11] H.-0. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers ofScience, Springer-Verlag, New York, 1992. [12] A. B. Chhabra, C. Meneveau, R. V. Jensen, and K. R. Sreenivasan, Direct determination of the f(a) singularity spectrum and its application to fully developed turbulence, Phys. Rev. A, 40-9 (1989), 5284-5294. [13] T. C. Halsey, M. H. Jensen, L. P. Kadanoff,l. Procaccia, and B.l. Shraiman, Fraetat measures and their singularities, Phys. Rev. A, 33 (1986), 1141.

20 Controlling Chaos

Aims and Objectives • To provide a brief historical introduction to chaos control. • To introduce two methods of chaos control for one- and two-dimensional discrete maps. On completion of this chapter, the reader should be able to • control chaos in the logistic and Henon maps; • plot time series data to illustrate the control; • appreciate how chaos control is being applied in the real world. This chapter is intended to give the reader a briefintroduction into the new and exciting field of chaos control and to show how some of the theory is being applied to physical systems. There has been considerable research effort into chaos control over the past decade, and practical methods have been applied in, for example, biochemistry, cardiology, communications, physics laboratories, and turbulence. Chaos control has been achieved using many different methods, but this chapter will concentrate on two procedures only. lt is possible to control chaos in both continuous and discrete nonlinear dynamical systems. However, our analysis will be restricted to discrete systems in this chapter.

330

20. Controlling Chaos

20.1

Historical Background

Even simple, well-defined continuous and discrete nonlinear dynamical systems without random terms can display highly complex, seemingly random behavior. Some of these systems have been investigated in this book, and mathematicians have Iabeted this phenomenon determinisitc chaos. Nondeterministic chaos, where the underlying equations are not known, such as that observed in a lottery or on a roulette wheel, will not be discussed in this text. Throughout history, dynamical systems have been used to model both the natural and technological sciences. In the early years of investigations, deterministic chaos was nearly always attributed to random external influences and was designed out if possible. The French mathematician and philosopher Henri Poincare laid down the foundations of the qualitative theory of dynamical systems at the turn of the century and is regarded by many as being the first chaologist. Poincare devoted much ofhis life in attempting to determine whether or not the solar system is stable. Despite knowing the exact form of the equations defining the motions of just three celestial bodies, he could not always predict the long-term future of the system. In fact, it was Poincare who first introduced the notion of sensitivity to initial conditions and long-term unpredictability. In recent years, deterministic chaos has been observed when applying simple models to cardiology, chemical reactions, electronic circuits, Iaser technology, population dynamics, turbulence, and weather forecasting. In the past, scientists have attempted to remove the chaos when applying the theory to physical models, and it is only in the last decade that they have come to realize the potential uses for systems displaying chaotic phenomena. For some systems, scientists are beginning to replace the maxim "stability good, chaos bad" with "stability good, chaos better." lt has been found that the existence of chaotic behavior may even be desirable for certain systems. Since the publication of the seminal paper of Ott, Grebogi, and Yorke [12] in 1990, there has been a great deal of progress in the development of techniques for the control of chaotic phenomena. Basic methods of Controlling chaos along with several reprints of fundamental contributions to this topic may be found in the excellent textbook ofKapitaniak [4]. Some ofthese methods will now be discussed very briefly, and then a selection of early applications of chaos control in the real world will be Iisted. I. Changing the system parameters. The simplest way to suppress chaos is to

change the system parameters in such a way as to produce the desired result. In this respect, bifurcation diagrams can be used to determine the parameter values. For example, in Chapter 17, bifurcation diagrams were used to determine regions of bistability for nonlinear bistable optical resonators. It was found that isolated bistable regions existed for only a narrow range of parameter values. However, the major drawback with this procedure is that large parameter variations may be required, which could mean redesigning

20.1. Historical Background

331

the apparatus and changing the dimensions of the physical system. In many practical situations, such changes are highly undesirable. II. Applying a damper. A common method for suppressing chaotic oscillations is to apply some kind of damper to the system. In mechanical systems, this would be a shock absorber, and for electronic systems, one might use a shunt capacitor. Once more, this method would mean a large change to the physical system and might not be practical. III. Pyragas :S method. This method can be divided into two feedback-controlling mechanisms: linear feedback control and time-delay feedback control. In the first case, a periodic extemal force is applied whose period is equal to the period of one of the unstable periodic orbits contained in the chaotic attractor. In the second case, self-controlling delayed feedback is used in a similar manner. This method has been very successful in controlling chaos in electronic circuits such as the Duffingsystem and Chua's circuit. A simple linear feedback method has been applied to the logistic map in Section 20.2. IV. Stabilizing unstable periodic orbits (the OGY method). The method relies on the fact the chaotic attractors contain an infinite number of unstable periodic orbits. By making small time-rlependent perturbations to a control parameter, it is possible to stabilize one or more of the unstable periodic orbits. The method has been very successful in applications, but there are some drawbacks. This method will be discussed in some detail at the end of this section. V. Occasional proportionalfeedbac k (OPF). Developed by Hunt [7] in 1991, this is one of the most promising control techniques for real applications. lt is a one-dimensional version of the OGY method and has been successful in suppressing chaos for many physical systems. The feedback consists of a series of kicks, whose amplitude is determined from the difference of the chaotic output signal from a relaxation oscillation embedded in the signal, applied to the input signal at periodic intervals. VI. Synchronization. The possibility of synchronization of two chaotic systems was first proposed by Pecorra and Carroll [11] in 1990 with applications in communications. By feeding the output from one chaotic oscillator (the transmitter) into another chaotic oscillator (the receiver), they were able to synchronize certain chaotic systems for certain parameter choices. This method opens up the possibilities for secure information transmission. Before summarizing the OGY method, it is worthwhile to highlight some of the other major results not mentioned above. The first experimental suppression of chaos was performed by Ditto, Rausseo, and Spano [9] using the OGY algorithm. By making small adjustments to the amplitude of an extemal magnetic field, they

332

20. Controlling Chaos

were able to stabilize a gravitationally buckled magnetostrictive ribbon that oscillated chaotically in a magnetic field. They produced period-one and period-two behavior, and the procedure proved tobe remarkably robust. Using both experimental and theoretical results, Singer, Wang, and Bau [8] applied a simple on-off strategy to laminarize (suppress) chaotic ftow of a fluid in a thermal convection loop. The on-off controllerwas applied to the Lorenz equations, and the numerical results were in good agreement with the experimental results. Shortly afterwards, Hunt [7] applied a modified version of the OGY algorithm called occasional proportional feedback (OPF) to the chaotic dynamics of a nonlinear diode resonator. Small perturbations were used to stabilize orbits of low period, but larger perturbations were required to stabilize orbits ofhigh periods. By changing the Ievel, width, and gain of the feedback signal, Hunt was able to stabilize orbits with periods as high as 23. Using the OPF algorithm developed by Hunt, Roy et al. [6] were able to stabilize a weakly chaotic green Iaser. In recent years, the implementation of the control algorithm has been carried out electronically using eilher digital signals or analog hardware. The hope for the future is that all-optical processors and feedback can be used to increase speed. The first experimental control in a biological systemwas performed by Garfinkel et al. [5] in 1992. They were able to stabilize arrhythmic behavior in eight out of 11 rabbit hearts using a feedback-control mechanism. lt was reported in [2] that a company has been set up to manufacture small defibrillators that can monitor the heart and deliver tiny electrical pulses to move the heart away from fibrillation and back to normality. It was also conjectured in the same article that the chaotic heart is more healthy than a regularly beating periodic heart. The OGY algorithm was implemented theoretically by the author and Steele [I] to control the chaos within a hysteresis cycle of a nonlinear bistable optical resonator using the real and imaginary parts of the electrical field amplitude. The same authors have recently managed to control chaos theoretically using the input intensity alone. This quantity is easy to continuously monitor and measure and should Iead to physical applications in the future. Methods 1-VI and the results given above are by no means exhaustive. This section is intended to provide a brief introduction to the subject and to encourage further reading. The Ott, Grebogi, and Yorke (OGY) Method. Consider the n-dimensional map (20.1)

Zn+ I = f(Zn, p),

where p is some accessible system parameter that can be changed in a small neighborhood of its nominal value, say po. In the case of continuous-time systems, such a map can be constructed by introducing a transversal surface of section and setting up a Poincare map. It is weil known that a chaotic attractor is densely filled with unstable periodic orbits and that ergodicity guarantees that any small region on the chaotic attractor will be visited by a chaotic orbit. The OGY method hinges on the existence of

20.1. Historical Background

333

stable manifolds around unstable periodic points. The basic idea is to make small time-dependent linear perturbations to the control parameter p in order to nudge the state towards the stable manifold of the desired fixed point. Note that this can only be achieved if the orbit is in a small neighborhood, or control region, of the fixed point. Suppose that Zs(P) is an unstable fixed point of equation (20.1). The position of this fixed point moves smoothly as the parameter p is varied. For values of p close to PO in a small neighborhood of Zs(po). the map can be approximated by a linear map given by (20.2)

Zn+l- Zs(Po) = J(Zn- Zs(Po))

g!.

+ C(p- Po),

where J is the Jacobian and C = All partial derivatives are evaluated at Zs(po) and PO· Assurne that in a small neighborhood around the fixed point (20.3)

P -Po = -K(Zn - Zs(po)),

where K is a constant vector of dimension n to be determined. Substitute (20.3) into (20.2) to obtain (20.4)

Zn+ I - Zs(po) =

(J- CK)(Zn- Zs(po)).

The fixed point is then stable as long as the eigenvalues, or regulator poles, have modulus less than unity. The pole-placement technique from control theory can be applied to find the vector K. A specific example is given in Section 20.3. A simple schematic diagram is given in Figure 20.1 to demoostrate the action of the OGY algorithm. Physically, one can think of a marble placed on a saddle: If the marble is rolled towards the center (where the fixed point lies), then it will roll off, as depicted in Figure 20.1 (a). However, if the saddle is moved slightly from side to side, say, by applying small perturbations, then the marble can be made to balance at the center ofthe saddle, as depicted in Figure 20.l(b). The following are some useful points to note: • The OGY technique is a feedback-control method. • Ifthe equations are unknown, sometimes delay-coordinate-embedding techniques using a single variable time series can be used. (The map can be constructed from experimental data.) • There may be more than one control parameter available. • Noise may affect the control algorithm. If the noise is relatively small, the control algorithm will still work in general. It should also be pointed out that the OGY algorithm can be applied only after the orbit has enteredasmall control region around the fixed point. For certain nonlinear systems, the number of iterations required-and hence the time-for the orbit to enter this control region may be too many tobe practical. Shinbrot et al. [ l 0] solved

334

20. Controlling Chaos

! (a)

(b)

Figure 20.1: Possible iterations near the fixed point (a) without control and (b) with control. The double-ended arrows are supposed to represent small perturbations to the system dynamics. The iterates Zj represent perturbed orbits. this problern by targeting trajectories to the desired control regions in only a small number of iterations. The method has also been successfully applied in physical systems.

20.2

Controlling Chaos in the Logistic Map

Consider the logistic map given by (20.5) as introduced in Chapter 14. There are many methods available to control the chaos in this one-dimensional system, but the analysis is restricted to periodic proportional pulses in this section. For more details on the method and its application to the Henon map, the reader is directed to [3]. To control chaos in this system, instantaneous pulses will be applied to the system variables Xn once every p iterations suchthat x; ｾ＠

kx;,

where k is a constant to be determined and p denotes the period. Recall that a fixed point of period one, say xs, of equation (20.5) satisfies the equation xs = f/J.(xs).

and this fixed point is stable if and only if

20.2. Controlling Chaos in the Logistic Map

335

Define the composite function F"(x) by

F"(x) = kf$(x). A fixed point of the function F" satisfies the equation (20.6)

kf$(xs) = xs,

where the fixed point xs is stable if (20.7)

lk

､ｦｾｸｳＩ＠

I

< I.

Define the function CP(x) by

CP(x} = _x_df$(x s). J$(x) dx Substituting from (20.6), equation (20.7) becomes (20.8) A fixed point of this composite map is a stable point of period p for the original logistic map when the control is switched on, providing condition (20.8) holds. In practice, chaos control always deals with periodic orbits oflow periods, say p = 1 to 4, and this method can be easily applied. To illustrate the method, consider the logistic map when 1-L = 4 and the system is chaotic. The functions C 1(x), C 2(x), C 3 (x), and C 4 (x) are shown in Figure 20.2. Figure 20.2(a) shows that fixed points of period one can be stabilized for every xs in the range between zero and approximately 0.67. When p = 2, Figure 20.2(b) shows that fixed points of period two can be stabilized only in three ranges of xs values. Figures 20.2(c) and (d) indicate that there are seven and 14 acceptable ranges for fixed points of periods three and four, respectively. Notice that the control ranges are getting smaller and smaller as the periodicity increases. Figure 20.3 shows time series data for specific examples when the chaos is controlled to period-one, period-two, period-three, and period-four behavior, respectively. The values of xs chosen in Figure 20.3 were derived from Figure 20.2. The values of k were calculated using equation (20.6). Note that the system can be stabilized to many different points on and even off the chaotic attractor (see Chau [3]). A Maple program is Iisted in Section 20.4. This method of chaos control by periodic proportional pulses can also be applied to the two-dimensional discrete Henon map. The interested reader is again directed to [3]. The OGY algorithm will be applied to the Henon map in the next section.

20. Controlling Chaos

336

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

CO

C' 0

...{),2 ...{),4

-0.6

-I

0.2

0.4

X

0.6

-I

0.8

(a)

0.2

0.4

X

0.6

0.8

(b)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

c' o -0.2

ＭｉＮｌｬ｟ｾｊＬＱ＠

0.2

0.4

(c)

X

0.6

0.8

ＭｬＮｌｯＬｊＲＫＴ｟Ｖｾｳ＠ X

(d)

Figure 20.2: Control curves Ci, i = I, 2, 3, 4, for the logistic map when J.t The range is restricted to -I < CP(xs) < 1 in each case.

20.3

= 4.

Controlling Chaos in the Henon Map

Ott, Grebogi, and Yorke [12] used the Henon map to illustrate the control method. A simple example will be given here. Consider the Henon map as introduced in

337

20.3. Controlling Chaos in the Henon Map

0.8

0.6 0.6

x.

x,

0.4 0.4

0.2 0.2

0 n n

(a)

(b)

0.8

0.6

x.

x.

0.4

0.2

0 n

(c)

(d)

Figure 20.3: Stabilization of points of periods one, two, three, and four for the logistic map when JL = 4: (a) xs = 0.4, k = 0.417; (b) xs = 0.2, k = 0.217; (c) xs = 0.3, k = 0.302; (d) xs = 0.6, k = 0.601. In each case, k is computed to three decimal places.

20. Controlling Chaos

338

Chapter 14. The two-dimensiona1 iterated map function is given by (20.9) where a > 0 and lßl < l. Take a transformation Xn system (20.9) becomes (20.10)

Xn+J-CX+

ßYn-Xn,2

Yn+l

= ｾｘｮ＠

and Yn

= ｾｙｮﾷ＠

Then

= Xn.

The proofthat system (20.9) can be transformed into system (20.10) will be left to the reader in the exercises at the end of the chapter (Section 20.5). The Henon map is now in the form considered in [12], and the control algorithm given in Section 20.1 will now be applied to this map. Set ß = 0.4 and allow the control parameter, in this case a, to vary araund a nominal value, say ao = 1.2, for which the map has a chaotic attractor. The fixed points of period one are determined by solving the simultaneaus equations ao + ßy - x 2 - x = 0 and x - y = 0. In Chapter 14, it was shown that the Henon map has two fixed points of period one if and only if (1 - ß) 2 +4ao > 0. In this particular case, the fixed points of period one are located approximately at A = (xl.l, YI.I) = (0.8357816692, 0.8357816692) and B = (XJ,2, Yl.2) = (-1.435781669 , -1.435781669) . The chaotic attractor and points of period one are shown in Figure 20.4. 2

y 0

ｾｌＭｯ｣Ｒ＠ X

Figure 20.4: Iterative plot for the Henon map (3000 iterations) when ao and ß = 0.4. The two fixed points ofperiod one are Iabeted A and B.

= 1.2

339

20.3. Controlling Chaos in the Henon Map The Jacobian matrix of partial derivatives of the map is given by J=

where P(x, y)

(

ｾ＠

iJP iJx

iJP ) iJy

!!.Q

iJx

'

iJy

= ao + ßy- x 2 and Q(x, y) = x. Thus -2x J = ( 1

ß ) 0 .

Consider the fixed point at A; the fixed point is a saddle point. Using the notation introduced in Section 20.1, for values of a close to ao in a small neighborhood of A, the map can be approximated by a linear map (20.11)

Zn+ I

-

Zs(ao) = J(Zn- Zs(ao)) + C(a- ao),

where Zn= (xn, Yn)T, A = Zs(ao), J is the Jacobian, and

C=

(

ｾ＠

iJP )

,

and allpartial derivatives are evaluated at ao and Zs(ao). Assurne that in a small neighborhood of A,

a- ao

(20.12)

= -K(Zn- Zs(ao)),

where

Substitute (20.12) into (20.11) to obtain

Zn+ I

-

Zs(ao) = (J- CK)(Zn - Zs(ao)).

Therefore, the fixed point at A = Zs(ao) is stable if the matrix J - CK has eigenvalues (or regulator poles) with modulus less than unity. In this particular case,

J _ CK ｾ＠

( -1.67156:338- k1

0.4 ｾ＠

k2

) '

and the characteristic polynomial is given by

Suppose that the eigenvalues (regulator poles) are given by Ai and

>..2. Then

340

20. Controlling Chaos

The lines of marginal stability are determined by solving the equations AJ = ±1 and AJA2 = 1. These conditions guarantee that the eigenvalues Al and J..2 have modulus less than unity. Suppose that AJA2 = 1. Then

Suppose that A.t =

+ 1. Then

>..2 = k2- 0.4

and

J..2 = -2.671563338- kJ.

Therefore,

k2 = -k) -2.271563338. If At = -1, then A.z

= -(k2- 0.4)

and

.l..2 = -0.671563338- kJ.

Therefore,

k2 = k) + 1.071563338. The stable eigenvalues (regulator poles) lie within a triangular region as depicted in Figure 20.5.

-4

-3

-2

Figure 20.5: The bounded region where the regulator poles are stable. Select kt = -1.5 and kz = 0.5. This pointlies well inside the triangular region shown in Figure 20.5. The perturbed Henon map becomes (20.13) Xn+l = ( -k] (Xn -

XJ,I)-

k2(Yn- Yt.d

+ ao) + ßYn -

x;,

Yn+l = Xn.

20.4. Maple Commands

341

Applying equations (20.10) and (20.13) without and with control, respectively, it is possible to plot time series data for these maps. Figure 20.6(a) shows a time series plot when the control is switched on after the 200th iterate; the control is left switched on until the 500th iterate. In Figure 20.6(b), the control is switched on after the 200th iterate and then switched off after the 300th iterate. 6·

4

r3 3

0

r}

I00

200

n

300

400

(a)

(b)

Figure 20.6: Time series data for the Henon map with and without control; r 2 = + y 2• In case (a), the control is activated after the 200th iterate, and in case (b), the control is switched off after the 300th iterate.

x2

Once again, the Maple program is listed in Section 20.4.

20.4

Maple Commands

># Controlling chaos in the logistic map # >######################################### > ># Define the logistic function and find f(f(x)) . >mu:=4 : >f:=x->mu*x*(l-x) : >ff:=f(f(x));

ff

:= 16x(l - x)(l - 4x(1 - x))

># Find k when xs=0.2: >0 . 2/f(f(0.2));

0.2170138888

342

20. Controlling Chaos

># Initialize. > x:=array(0 .. 10000): > x(0]:=0.6:imax:=l00:k:=0.217: >

># Switch on the control after the 60th iterate. ># Kick the system every second iterate. > for i from 0 by 2 to imax do > x[i+l):=mu*x[i)*(l-x[i)l:x[i+2):=mu*x[i+1]*(1-x[i+l]l: > if i>SO then > x[i+l]:=k*mu*x(i]*(1-x[i]):x[i+2):=mu*x[i+1)*(1-x[i+1]):fi:od: >

># Plot the time series data. > wi th (plots l : > pts:=[[m,x(m]]$m=O .. imax]: > p1:=plot(pts,style=point,symbol=circle,color=black): > p2:=plot{pts,x=O .. imax,y=0 .. 1,color=black): > disp1ay{{p1,p2},1abe1s=['','')); >#See Figure 20.3(b). >

># Controlling chaos in the Henon map # >###################################### >

>x:=array(O .. l0000):y:=array(0 .. 10000) :rsqr:=array(0 .. 10000): >alpha:=l.2:beta:=0.4: >

># Find the fixed points of period one. ＾ｳｯｬｶ･ｻ｡ｰｨＭｸｾＲＫ｢ｴＪｹＬｽＩ［＠

{y=0.8357816692,x=0.8357816692},{x=-1.435781669,y=-1.435781669} >xstar:=0.8357816692:ystar:=xstar: >with(lina1g): >A:=matrix([[-2*xstar-kl,beta-k2], [1,0]));

A := [ ＭＱＮＶＷＵｾＳＸ

kl

0.4 ｾ＠

k2 ]

># Determine the characteristic polynomial. >expand((-1.671563338-k1-lambda)*(-lambdal-(beta-k2));

A. 2 + (1.671563338 + kl)A. + (k2- 0.4) ># Iterate the system and switch on the control after 100 ># and iterations. In this case, the regulator poles are kl=-1.8 ># k2=1. 2. >x(O} :=0.5:y[0]:=0.6:imax:=499: >kl:=-1.8:k2:=1.2: >

>for i from 0 to imax do ＾ｸ｛ｩＫＱ｝Ｚ］｡ｬｰｨ｢･ｴＪｹＭＨＩｾＲ＠

>y[i+l] :=x(i]: >if i>200 then

20.5. Exercises

343

>x[i+l) :=(-kl*(x [i)-xstar) -k2*(y[i) -ystar)+a lpha)+be ta*y[i) ＾ＭＨｸ｛ｩｬｾＲＺ＠

>y [ i + 1] : =x [ i 1 : >fi:

>od: >

># Determine the square of the distance of each point from the ># origin. >for j from 0 to imax do >rsqr [ j 1 : =evalf ( (x [ j 1l ｾＲＫ＠ (y [ j Jl A2) : >od: >

>wi th (plots l : >points:= [[m,rsqr(m ])$m=O .. imax]: >plot({po ints},x=O .. imax,y=0 .. 6); ># See Figure 20.6(a).

20.5

Exercises

1. Show that the map defined by Xn+l = 1 + Yn- ax;,

Yn+l = bxn

can be written as Un+l = a

+ bVn- ｕｾＬ＠

Vn+l = Un

using a suitable transformation. 2. Apply the method of chaos control by periodic proportional pulses (see Section 20.2) to the logistic map Xn+l

= J.l.Xn(l -

Xn)

when J.l. = 3.7. Sketch the graphs Ci (x), i = 1 to 4. Plot time series data to illustrate control of fixed points of periods one, two, three, and four. 3. Find the points of periods one and two for the Henon map given by Xn+l = a

when a

+ byn- x;,

Yn+l = Xn

= 1.4 and b = 0.4, and determine their type.

4. Apply the method of chaos control by periodic proportional pulses (see Section 20.2) to the two-dimensional Henon map Xn+l

= a + byn- x;,

Yn+l

= Xn,

where a = 1.4 and b = 0.4. (In this case, you must multiply Xm by k1 and Ym by k2, say, once every p iterations). Plot time series data to illustrate the control of points of periods one, two, and three.

20. Controlling Chaos

344

5. Use the OGY algorithm given in Section 20.3 to stabilize a point of period one in the Henon map Xn+l = a when a

+ byn- x;,

Yn+l = Xn

= 1.4 and b = 0.4. Display the control using a time series graph.

6. Consider the Ikeda map (introduced in Chapter 16) given by

Suppose that En = Xn map in Xn and Yn·

+ iYn· Rewrite the Ikeda map as a two-dimensional

7. Plot the chaotic attractor for the Ikeda map

when A = 2.7 and B = 0.15. How many points are there of period one? Indicate where these points are with respect to the attractor. 8. Plot the chaotic attractor for the Ikeda map

when (i) A

= 4 and B = 0.15;

(ii) A

= 7 and B = 0.15.

How many points are there of period one in each case? Indicate where these points are for each of the attractors on the figures. 9. Use the OGY method (see Section 20.3) with the parameter A to control the chaos to a point of period one in the Ikeda map

when Ao = 2.7 and B = 0.15. Display the control on a time series p1ot. (N.B.: Use a two-dimensional map.) 10. Try the same procedure of control to period one for the Ikeda map as in exercise 9 but with the parameters Ao = 7 and B = 0.15. Investigate the size of the control region araund one of the fixed points in this case and state how it compares to the control region in exercise 9. What can you say about flexibility and controllabi1ity?

20.5. Exercises

345

Recommended Reading [1] S. Lynch and A. L. Steele, Controllingchaos in nonlinear bistable optical resonators, Chaos, Solitons, Fractals, 11-5 (2000), 721-728. [2] M. Buchanan, Fascinating rhythm, New Scientist, 3 January, 1998, 20-25. [3] N. P. Chau, Controllingchaos by periodic proportional pulses, Phys. Lett. A, 234 (1997), 193-197. [4] T. Kapitaniak, Controlling Chaos: Theoretical and Practical Methods in Non-Linear Dynamics, Academic Press, New York, 1996. [5] A. Garfinkel, M. L. Spano, W. L. Ditto, and J. N. Weiss, Controlling cardiac chaos, Science, 257 (1992), 1230-1235. [6] R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, Dynamical control of a chaotic Iaser: Experimental stabilization of a globally coupled system, Phys. Rev. Lett., 68 (1992), 1259-1262. [7] E. R. Hunt, Stabilizing high-period orbits in a chaotic system: The diode resonator, Phys. Rev. Lett., 67 (1991), 1953-1955. [8] J. Singer, Y.-Z. Wang, and H. H. Bau, Controlling a chaotic system, Phys. Rev. Lett., 66 (1991), 1123-1125. [9] W. L. Ditto, S. N. Rausseo, and M. L. Spano, Experimental control of chaos, Phys. Rev. Lett., 65 (1990), 3211-3214. [10] T. Shinbrot, C. Grebogi, E. Ott, and J. A. Yorke, Using chaos to direct trajectories to targets, Phys. Rev. Lett., 65 (1990), 3215-3218. [11] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824. [12] E. Ott, C. Grebogi, and J. A. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), 1196-1199.

21 Examination-Type Questions

21.1 1.

Dynamical Systems with Applications (a) Sketch phase portraits for the following systems: (i) x = y- 3x, y = -2y; (ii) x = y, y = x2 .

(b) Sketch a phase portrait in three-dimensional space for the linear system

x=x-2y, 2.

y=2x+y.

z=4z.

(a) Sketch a phase portrait for the system

x = x(4- y- x),

y=

y(3x- 1- y), ｘｾ＠

0,

y

0 ｾ＠

given that the critical points occur at 0 = (0, 0), A = (4, 0), and

B=(i.lj).

(b) Sketch a phase portrait for the system x = x(2- y- x),

y = y(3- 2x- y), ｘｾ＠

0,

y ｾ＠

0

given that the critical points occur at 0 = (0, 0), C = {0, 3), D = (2, 0), and E = (1, 1). One of the systems can be used to model predator-prey interactions and the other to model competing species. Describe which system applies to which model and interpret the results in terms of species behavior.

21. Examination·Type Questions

348 3.

(a) Prove that the system

x = y+ x(a- x 2 -i), y =

-x

+ y(l -

x 2 - y 2),

where 0 < a < 1, has a Iimit cycle and determine its stability. (b) Prove that none of the following systems has a Iimit cycle:

= -x + y 3 - y4, y = I - 2y- x 2y + x 4 ; x = x2 - y- 1, y = y(x- 2); x = x - y2(l + x 3 ), y = xs - y; x = 4x- 2x 2 - y 2, y = x(l +xy).

(i) x

(ii) (iii)

(iv) 4.

(a) Prove that the origin of the system

x = -x-xl-x 3 , y = -1y+3x 2y-2yz 2 -y3, z= -5z+y 2z-z 3 is globally asymptotically stable using a suitable Lyapunov function. (b) Sketch a phase portrait for the system

x = Y(i- 1), y = x(l- x 2). 5.

(a) Define the tent map T : [0, 1] --+- [0, 1] by T( ) X

=

{

7

4x 1(] -X) 4

I 0 :;: x < 2• = 2B 1 + B - lzsl 2 '

where zs is a fixed point of (21.1). 8.

(a) A variation ofthe Koch snowtlake is constructed by attaching the Koch curve to the outer edges of a unit square. The motif for the Koch curve is shown below.

Determine the area bounded by the true fractal if it is constructed to stage infinity. Find the fractal dimension of the object. (b) A certain species of insect can be divided into three age classes: 0-6 months, 6-12 months, and 12-18 months. A Leslie matrix for the female population is given by

Determine the Iong-term distribution of the insect population when (i) no insecticide is applied; (ii) an insecticide is applied that kills 10% of the youngest age class, 20% of the middle age class, and I 0% of the oldest age class in each six-month period.

350

21. Examination·Type Questions

21.2 1.

Dynamical Systems with Applications Using Maple (a) The logistic map is defined by Xn+l = /p.(Xn) = J.LXn(l - Xn),

where 0 ｾ＠ Xn ｾ＠ 1. Compute /;(x), JJ(x), and f:(x). Suppose that J.L = 3.9. Plot graphs of fp., and Estimate the fixed points, if any, of periods one, two, three, and four, respectively.

1:. JJ, 1:.

(b) Prove that J.L = 1 + -/6 is a bifurcation point for the logistic map. 2. A predator-prey system can be modeled using the differential equations x

= x(l -

y = y( -1 + x -

y - u),

E"y),

where x(t) is the population ofprey and y(t) is the predator population size at time t, respectively. Classify the critical points for E ｾ＠ 0 and plot phase portraits for the different types of qualitative behavior. Interpret the results in physical terms. 3.

(a) By plotting suitable graphs, show that the system . 2 X. =X - y - 2X SlO

(1rX) 2 ,

has a unique critical point inside the square, say D, oflength 2 centered at the origin. Prove that there exists a Iimit cycle inside D using the corollary to the Poincar6-Bendixson Theorem. Plot a phase portrait for this system. (b) Using Bendixson 's criterion with a multiplier 1/f = x 0 yh, where a and b are constants, prove that the systern

x= x -

4x 2 + xy,

y=

2y + 3xy -

2i

has no Iimit cycles. 4.

(a) A certain species of fish can be divided into four age classes, each six months long. The Leslie rnodel forthefemale portion of the population is given by 0 2 2 0 4 L = ( ｾ＠

0 3' 0 0

H). i

0

21.2. Dynamical Systems with Applic:ations Using Maple

351

lf the initial population is given by

Xo= (

ｾＩＮ＠

deterrnine the population after six, 18, and 30 months, respectively. Show that the population eventually increases by 57.57% every six months, and deterrnine the long-terrn age distribution if no culling takes place. The species of fish is to be harvested using one of five different policies: either harvest Ｑ ｾ＠ of each of the four age classes in turn or harvest ｾ＠ of each age class at the end of each six-month period. Deterrnine the rate of population increase/decrease and the long-terrn age distribution in each of the five cases. (b) lf possible, what proportion of the population would have tobe harvested to produce a sustainable policy if (i) harvesting is uniform across the four age classes, or (ii) only the youngest age class is harvested? 5.

(a) A multifractal Cantor set is constructed using the motif

Po= 1

Stage 0

where2pJ +P2 = l.Atstagek,eachsegmentisoflength k and there are N = 3k segments. If PI = and P2 = deterrnine expressions foras and fs and hence plot an f(a) curve.

!

!.

(b) Estimate the fixed points of periods one and two for the Henon map given by Xn+1 = 1- 1.2x; + Yn. Yn+1 = 0.4xn. Plot the chaotic attractor.

22 Salutions to Exercises

22.0

Chapter 0

Edit the Maple commands given in this chapter. 1.

(a) 11;

(b) 1024; (c) 0.09983341665;

(d) -40.

2.

(a) 9x 2 + 4x; (b)

2x3

Jt+x 4

;

(c) ex(sin(x) cos(x) 3.

(a) ＭＱｾ［＠

(b) 1; (c)

4.

..;'1i.

(a) 1; (b)

!;

(c) 0.

+ cos 2(x)- sin 2 (x)).

22. Solutions to Exercises

354

5.

(a) -I+ 61; (b) -

?o + 19o1 ;

+ 2.287355287 I; 1.282474679 + 0.9827937233/; -2.8472390 87 + 2.370674169/.

(c) 1.468693940

(d) (e)

6.

ＲａＭｂｃ］ＨＡｾ＠

(a)

-2 B- 1 = (

(b)

ｾ＠ ｾＲ＠

-1

］ｾ＠ ］ｾＩ［＠ I

-7

］ｾ＠ I

);

(c) Eigenvectors of C are [3, I, [-1, I, -2]],

[0, I, [-1, I, 1]],

[2, 1[1, -2, 2]].

7. (d)

4 Figure 22.1: A three-dimensional plot of the function z = 4x 2eY - 2x 4 - e Y. The surface has two local maxima but no local minima. You can rotate the figure by clicking on it with the left mouse button.

8.

+ 2; -2e- 31 + 3e- 21 .

(a) y(x) = iJ2x2 (b) x(t) =

355

22.1. Chapter 1

9.

(a) When x[O] :=0.2,

x[91] :=0.8779563852 x[92] :=0.4285958836 x[93] :=0.9796058084 x[94] :=0.7991307420e-1 x[95] :=0.2941078991 x[96] :=0.8304337709 x[97]:=0.5632540923 x[98]:=0.9839956791 x[99]:=0.6299273044e-1 x[lOO] :=0.2360985855 (b) When x[O): =0. 2001,

x[91):=0.6932414820 x[92) :=0.8506309185 x[93) :=0.5082318360 x[94] :=0.9997289475 x[95] :=0.1083916122e-2 x[96) :=0.4330964991e-2 x[97]:=0.1724883093e-1 x[98) :=0.6780523505e-1 x[99]:=0.2528307406 x[lOO] :=0.7556294285 10. > # Euclid' s algorithm >

> > > > >

a:=l2348:b:=14238: while bO do d:=irem(a,b): a:=b:b:=d:od: lprint('The greatest common divisor is',a):

The greatest common divisor is 126.

22.1 1.

Chapter 1 (a) Y =

f;

(b) y = Cx 2 ; (c) y = C,Ji;

(d) ｾ＠ = ln(f); (e) ｾ＠

4

+ !:F = C;

(t) y =

2 2

I

ce-x.

22. Solutions to Exercises

356 2. The fossil is 8.03 x 106 years old.

kJ(ao- d)(bo- d)(co- d)- k,(do + d); (b) .X = kf(ao- 3x) 3 - k,x, where a = [A], x = [A3], b = [B], c = [C], and d = [D]. (a)

3.

d=

(a) The current is I = 0.733A;

4.

(b) the charge is Q(t) = 50(1 - exp( -lOt - t 2 )) coulombs. 5. Time= 1.18 hours. 6. The concentration of glucose is

G

g(t) = - -

lOOk V

-ce- kt .

7. The differential equations are

Ä = -aA,

B=

aA- ßB,

C = ßB.

9. The differential equations are

i = aH- (b + c)l, b =

ii = -aH + b/,

cl.

The number of dead is given by D(t) = acN (

a-ß+ ßeat- aeß') , aß(a- ß}

where a and ß are the roots of >.. 2 + (a + b + c)>.. + ac = 0. This is not realistic as the whole population eventually dies. In reality, people recover and some become immune. 10.

(a) (i) The solution isx 3 = l/(l-3t} with maximal interval (MI) -oo < t < j; (ii) x(t) = (e' + 3}/(3 - e') with MI -oo < t < ln 3; (iii) x(t) = 6/(3- e 2') with MI -oo < t < ln ./3. (b) The solution is x(t) = (t

22.2 1.

+ xci 12 -

to) 2 with MI to- xci 12 < t < oo.

Chapter 2 (a) All trajectories are vertical and there are an infinite nurober of critical points on the line y = - ｾＮ＠ (b) All trajectories arehorizontal and there are an infinite number of critical points on the line y = - ｾＮ＠

22.3. Chapter 3

357

2. Eigenvalues and eigenvectors are Ä1 The origin is a stable node.

= -10, (-2, 1) 7 ; Ä2 = -3, ｻｾＬ＠

3. Eigenvalues and eigenvectors are Ä1 = -4, ( 1, 0) T; Ä2 = 2, (- ｾＬ＠ origin is a saddle point.

4.

1) 7 .

1) T. The

(a) Eigenvalues andeigenvec torsareÄ1 =5,(2, l) 7 ;Ä2 = -5,(1, -2) 7 . The origin is a saddle point. (b) Eigenvalues are Ä1 focus.

= 3 + i, Ä2 = 3- i, and the origin is an unstable

(c) There are two repeated eigenvalues and one linearly independent eigenvector: Ä1 = -1, ( -1, 1) 7 . The origin is a stable degenerate node. (d) This isanonsimp le fixed point. There are an infinite nurober of critical points on the line y = x. 5.

(a)

x = y, y = -25x -

(b)

(i) unstable focus,

J.LY;

(ii) centre, (iii) stable focus,

(iv) stable node; (c)

(i) oscillations grow, (ii) periodic oscillations, (iii) damping, (iv) critical damping. The constant J.L is called the damping coefficient.

22.3

Chapter 3

Edit the Maple commands in Section 3.3 to plot the relevant phase portraits.

1.

(a) There is one critical point at the origin which is a col. Plot the isoclines. The eigenvalues are Ä = -lj/5 with eigenvectors 1 ) and )

0, then the system has no Iimit cycles. If a1a3 < 0, there is a unique hyperbolic Iimit cycle. If a1 = 0 and a3 # 0, then there are no Iimit cycles. If a3 = 0 and a1 # 0, then there are no Iimit cycles. lf a1 = a3 = 0, then the origin is a center by the classical symmetry argument. 3. When E is small, one may apply the Melnikov theory of Chapter 10 to establish where the Iimit cycles occur. The Iimit cycles are asymptotic to circles centered at the origin. If the degree of F is 2m + I or 2m + 2, there can be no more than m Iimit cycles. When E is large, if a Iimit cycle exists, it shoots across in the horizontal direction to meet a branch ofthe curve y = F(x), where the trajectory slows down and remains near the branch until it shoots back across to another branch of F(x), where it slows down again. The trajectory follows this pattem forever. Once more, there can be no more than m Iimit cycles. 4. Use a similar argument to that used in Theorem 2. 5. The function F must satisfy the conditions a1 > 0, a3 < 0, and ai > 4aJ, for example. This guarantees that there are five roots for F (x ). If there is, say, a local maximum of F(x) at (aJ, 0), a root at (a2, 0), and a local minimum at (a3, 0), then it is possible to prove that there is a unique hyperbolic Iimit cycle crossing F(x) in the interval (aJ, a2) and a second hyperbolic Iimit cycle crossing F(x) in the interval (a3, oo). Again, similar arguments to those used in Theorem 2 can be applied.

22.13

Chapter 13

I. The general solution is Xn

2.

(a) Fn

(b)

(i)

(11") (iii)

= Ｒ ｮｾ｛Ｈｬ＠

= 7r(4n + cn(n -

+ y'j)n _

1)).

(1 _ y'j)n];

= ｾＲｮ＠ + j(-l)n- 1; 52n + 6I( - l)n - n - ｾ［＠ I Xn = 3 Xn = 2n + !en ( -l)n + jen2n Xn

!en.

22. Solutions to Exercises

368 3. The dominant eigenvalue is A. 1 = 1.107, and (a)

xos> = (

(b)

x =

(c)

X(IOO)

64932 ) ; 52799 38156

2.271 X 106 ( 1.847 x to6 1.335 X 1o6

)

3.645 X 108 2.964 X 108 2.142x toB

)

=(

;

•

= I and A2.3 = .::.!.p2. There is no dominant = IA.JI. The population stabilizes. eigenvalue since lA. 11 =

4. The eigenvalues are Al

IA.2I

5. The eigenvalues are 0, 0, -0.656 ± 0.626i, and A. 1 = 1.313. Therefore, the population increases by 31.3% every fifteen years. The normalized eigenvector is given by

0.415 0.283 0.173 0.092 0.035

x=

7. Before insecticide is applied, A.1 = 1.465, which means that the population increases by 46.5% every six months. The normalized eigenvector is

x= (

0.764) . o.2o8 0.028

=

After the insecticide is applied, A. 1 1.082, which means that the population increases by 8.2% every six months. The normalized eigenvector is given by

x= (

0.695) . 0.257 0.048

8. Forthis policy, d1 = 0.1, d2 = 0.4, and d3 = 0.6. The dominant eigenvalue is >.. 1 = 1.017 and the normalized eigenvector is

x= (

0.797 ) . o.t88 0.015

22.14. Chapter 14

369

9. Without any harvesting, the population would double each year since AJ = 2.

(i)

AI= 1,

(ii)

6 h) = -, 7

(iii)

AI = 1.558,

x=(!} x=(!}

x= c.780) 0.167 ; 0.053

(iv)

hi = 0.604,

(v)

AI= 1.433,

x=

AJ = 1.672,

x=

c.761) c.668) 0.111 0.062

;

0.132 0.199

.

10. Take h2 = h3 = 1. Then AI = 1, A2 = -1, and AJ = 0. The population stabilizes.

22.14 Chapter 14 I. The iterates give orbits with periods (i) one, (ii) one, (iii) three, and

(iv) nine. There are two points of period one, two points of period two, six points of period three, and 12 points of period four. In general, there are 2N -(sum of points of periods that divide N) points of period N. 2. The functions are given by 9 4x,

0 ｾｘ＼＠

ｾＭＪｘＬ＠

I

9

3

4x- 4• *(1- x),

t. I

:J ｾｘ＠

< !•

I 2" ｾｘ＼＠

3•

ｾｘ＠

l,

j

2

22. Solutions to Exercises

370 and

21x

0 ｾｘ＼Ｎ＠

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