Differential dynamical systems

435 Pages·2007·7.495 MB·English

by  Meiss J.D.

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Differential
Dynamical
Systems
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Mathematical Modeling
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Differential
Dynamical
Systems
James D. Meiss
University of Colorado
Boulder, Colorado
Society for Industrial and Applied Mathematics
Philadelphia
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Copyright © 2007 by the Society for Industrial and Applied Mathematics.
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All rights reserved. Printed in the United States of America. No part of this book may be
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Library of Congress Cataloging-in-Publication Data
Meiss, J. D.
Differential dynamical systems / James D. Meiss.
p. cm. — (Mathematical modeling and computation ; 14)
Includes bibliographical references and index.
ISBN 978-0-898716-35-1 (alk. paper)
1. Differential dynamical systems—Mathematical models. I. Title.
QA614.8.M45 2007
515’.39—dc22 2007061747
is a registered trademark.
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To Don and Peggy Meiss
for teaching me to explore,
and to Mary Sue Moore
for always believing that I would discover.
(cid:2)
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Contents
ListofFigures xi
Preface xvii
Acknowledgments xxi
1 Introduction 1
1.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 WhatAreDifferentialEquations? . . . . . . . . . . . . . . . . . . . . 2
1.3 One-DimensionalDynamics. . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
PopulationDynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 8
MechanicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
OscillatingCircuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
FluidMixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Two-DimensionalDynamics . . . . . . . . . . . . . . . . . . . . . . 14
Nullclines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
PhaseCurves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6 TheLorenzModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 QuadraticODEs: TheSimplestChaoticSystems . . . . . . . . . . . . 21
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 LinearSystems 29
2.1 MatrixODEs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . . . . . . 30
Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 Two-DimensionalLinearSystems . . . . . . . . . . . . . . . . . . . . 35
2.3 ExponentialsofOperators . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 FundamentalSolutionTheorem . . . . . . . . . . . . . . . . . . . . . 45
2.5 ComplexEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 MultipleEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Semisimple-NilpotentDecomposition. . . . . . . . . . . . . . . . . . 51
TheExponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
AlternativeMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
vii
viii Contents
2.7 LinearStability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.8 NonautonomousLinearSystemsandFloquetTheory. . . . . . . . . . 61
2.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 ExistenceandUniqueness 73
3.1 SetandTopologicalPreliminaries . . . . . . . . . . . . . . . . . . . . 73
Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
UniformConvergence . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2 FunctionSpacePreliminaries . . . . . . . . . . . . . . . . . . . . . . 76
MetricSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
ContractionMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
LipschitzFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 ExistenceandUniquenessTheorem . . . . . . . . . . . . . . . . . . . 84
3.4 DependenceonInitialConditionsandParameters . . . . . . . . . . . 92
3.5 MaximalIntervalofExistence. . . . . . . . . . . . . . . . . . . . . . 98
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4 DynamicalSystems 105
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 GlobalExistenceofSolutions . . . . . . . . . . . . . . . . . . . . . . 109
4.4 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.6 LyapunovFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.7 TopologicalConjugacyandEquivalence . . . . . . . . . . . . . . . . 130
4.8 Hartman–GrobmanTheorem . . . . . . . . . . . . . . . . . . . . . . 138
4.9 Omega-LimitSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.10 AttractorsandBasins . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.11 StabilityofPeriodicOrbits . . . . . . . . . . . . . . . . . . . . . . . 152
4.12 PoincaréMaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5 InvariantManifolds 165
5.1 StableandUnstableSets . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2 HeteroclinicOrbits. . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.3 StableManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.4 LocalStableManifoldTheorem . . . . . . . . . . . . . . . . . . . . . 173
5.5 GlobalStableManifolds . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.6 CenterManifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6 ThePhasePlane 197
6.1 NonhyperbolicEquilibriainthePlane . . . . . . . . . . . . . . . . . 197
6.2 TwoZeroEigenvaluesandNonhyperbolicNodes . . . . . . . . . . . 198
6.3 ImaginaryEigenvalues: TopologicalCenters . . . . . . . . . . . . . . 203
6.4 SymmetriesandReversors . . . . . . . . . . . . . . . . . . . . . . . 211
Contents ix
6.5 IndexTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
HigherDimensions: TheDegree . . . . . . . . . . . . . . . . . . . . 217
6.6 Poincaré–BendixsonTheorem . . . . . . . . . . . . . . . . . . . . . . 219
6.7 LiénardSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.8 BehavioratInfinity: ThePoincaréSphere . . . . . . . . . . . . . . . 229
6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7 ChaoticDynamics 243
7.1 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.2 LyapunovExponents . . . . . . . . . . . . . . . . . . . . . . . . . . 248
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
PropertiesofLyapunovExponents . . . . . . . . . . . . . . . . . . . 252
ComputingExponents . . . . . . . . . . . . . . . . . . . . . . . . . . 255
7.3 StrangeAttractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
HausdorffDimension . . . . . . . . . . . . . . . . . . . . . . . . . . 260
Strange,NonchaoticAttractors . . . . . . . . . . . . . . . . . . . . . 262
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
8 BifurcationTheory 267
8.1 BifurcationsofEquilibria . . . . . . . . . . . . . . . . . . . . . . . . 267
8.2 PreservationofEquilibria . . . . . . . . . . . . . . . . . . . . . . . . 271
8.3 UnfoldingVectorFields . . . . . . . . . . . . . . . . . . . . . . . . . 273
UnfoldingTwo-DimensionalLinearFlows . . . . . . . . . . . . . . . 275
8.4 Saddle-NodeBifurcationinOneDimension . . . . . . . . . . . . . . 278
8.5 NormalForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
HomologicalOperator . . . . . . . . . . . . . . . . . . . . . . . . . . 282
MatrixRepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Higher-OrderNormalForms . . . . . . . . . . . . . . . . . . . . . . 287
8.6 Saddle-NodeBifurcationinRn . . . . . . . . . . . . . . . . . . . . . 290
Transversality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
CenterManifoldMethods . . . . . . . . . . . . . . . . . . . . . . . . 293
8.7 DegenerateSaddle-NodeBifurcations . . . . . . . . . . . . . . . . . 295
8.8 Andronov–HopfBifurcation . . . . . . . . . . . . . . . . . . . . . . . 296
8.9 TheCuspBifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 301
8.10 Takens–BogdanovBifurcation . . . . . . . . . . . . . . . . . . . . . 304
8.11 HomoclinicBifurcations. . . . . . . . . . . . . . . . . . . . . . . . . 306
FragilityofHeteroclinicOrbits . . . . . . . . . . . . . . . . . . . . . 306
GenericHomoclinicBifurcationsinR2 . . . . . . . . . . . . . . . . . 309
8.12 Melnikov’sMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
8.13 Melnikov’sMethodforNonautonomousPerturbations . . . . . . . . . 314
8.14 ShilnikovBifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . 322
8.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
9 HamiltonianDynamics 333
9.1 ConservativeDynamics . . . . . . . . . . . . . . . . . . . . . . . . . 333
9.2 Volume-PreservingFlows . . . . . . . . . . . . . . . . . . . . . . . . 335