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Introduction 1

1 Equilibrium Solutions,Stability,and Linearized Stability 5

1.1 Equilibria of Vector Fields 5

1.2 Stability of Trajectories 7

1.2a Linearization 10

1.3 Maps 15

1.3a Definitions of Stability for Maps 15

1.3b Stability of Fixed Points of Linear Maps 15

1.3c Stability of Fixed Points of Maps via the Linear Approximation 15

1.4 Some Terminology Associated with Fixed Points 16

1.5 Application to the Unforced Duffing Oscillator 16

1.6 Exercises 16

2 Liapunov Functions 20

2.1 Exercises 25

3 Invariant Manifolds：Linear and Nonlinear Systems 28

3.1 Stable,Unstable,and Center Subspaces of Linear,Autonomous Vector Fields 29

3.1a Invariance of the Stable,Unstable,and Center Subspaces 32

3.1b Some Examples 33

3.2 Stable,Unstable,and Center Manifolds for Fixed Points of Nonlinear,Autonomous Vector Fields 37

3.2a Invariance of the Graph of a Function：Tangency of the Vector Field to the Graph 39

3.3 Maps 40

3.4 Some Examples 41

3.5 Existence of Invariant Manifolds：The Main Methods of Proof,and How They Work 43

3.5a Application of These Two Methods to a Concrete Example：Existence of the Unstable Manifold 45

3.6 Time-Dependent Hyperbolic Trajectories and their Stable and Unstable Manifolds 52

3.6a Hyperbolic Trajectories 53

3.6b Stable and Unstable Manifolds of Hyperbolic Trajectories 56

3.7 Invariant Manifolds in a Broader Context 59

3.8 Exercises 62

4 Periodic Orbits 71

4.1 Nonexistence of Periodic Orbits for Two-Dimensional,Autonomous Vector Fields 72

4.2 Further Remarks on Periodic Orbits 74

4.3 Exercises 76

5 Vector Fields Possessing an Integral 77

5.1 Vector Fields on Two-Manifolds Having an Integral 77

5.2 Two Degree-of-Freedom Hamiltonian Systems and Geometry 82

5.2a Dynamics on the Energy Surface 83

5.2b Dynamics on an Individual Torus 85

5.3 Exercises 85

6 Index Theory 87

6.1 Exercises 89

7 Some General Properties of Vector Fields：Existence,Uniqueness,Differentiability,and Flows 90

7.1 Existence,Uniqueness,Differentiability with Respect to Initial Conditions 90

7.2 Continuation of Solutions 91

7.3 Differentiability with Respect to Parameters 91

7.4 Autonomous Vector Fields 92

7.5 Nonautonomous Vector Fields 94

7.5a The Skew-Product Flow Approach 95

7.5b The Cocycle Approach 97

7.5c Dynamics Generated by a Bi-Infinite Sequence of Maps 97

7.6 Liouville's Theorem 99

7.6a Volume Preserving Vector Fields and the PoincaréRecurrence Theorem 101

7.7 Exercises 101

8 Asymptotic Behavior 104

8.1 The Asymptotic Behavior of Trajectories 104

8.2 Attracting Sets,Attractors,and Basins of Attraction 107

8.3 The LaSalle Invariance Principle 110

8.4 Attraction in Nonautonomous Systems 111

8.5 Exercises 114

9 The Poincaré-Bendixson Theorem 117

9.1 Exercises 121

10 Poincaré Maps 122

10.1 Case 1：Poincaré Map Near a Periodic Orbit 123

10.2 Case 2：The Poincaré Map of a Time-Periodic Ordinary Difierential Equation 127

10.2a Periodically Forced Linear Oscillators 128

10.3 Case 3：The Poincaré Map Near a Homoclinic Orbit 138

10.4 Case 4：Poincaré Map Associated with a Two Degree-of-Freedom Hamiltonian System 144

10.4a The Study of Coupled Oscillators via Circle Maps 146

10.5 Exercises 149

11 Conjugacies of Maps,and Varying the Cross-Section 151

11.1 Case 1：Poincaré Map Near a Periodic Orbit：Variation of the Cross-Section 154

11.2 Case 2：The Poincaré Map of a Time-Periodic Ordinary Differential Equation：Variation of the Cross-Section 155

12 Structural Stability,Genericity,and Transversality 157

12.1 Definitions of Structural Stability and Genericity 161

12.2 Transversality 165

12.3 Exercises 167

13 Lagrange's Equations 169

13.1 Generalized Coordinates 170

13.2 Derivation of Lagrange's Equations 172

13.2a The Kinetic Energy 175

13.3 The Energy Integral 176

13.4 Momentum Integrals 177

13.5 Hamilton's Equations 177

13.6 Cyclic Coordinates,Routh's Equations,and Reduction of the Number of Equations 178

13.7 Variational Methods 180

13.7a The Principle of Least Action 180

13.7b The Action Principle in Phase Space 182

13.7c Transformations that Preserve the Fcrm of Hamilton's Equations 184

13.7d Applications of Variational Methods 186

13.8 The Hamilton-Jacobi Equation 187

13.8a Applications of the Hamilton-Jacobi Equation 192

13.9 Exercises 192

14 Hamiltonian Vector Fields 197

14.1 Symplectic Forms 199

14.1a The Relationship Between Hamilton's Equations and the Symplectic Form 199

14.2 Poisson Brackets 200

14.2a Hamilton's Equations in Poisson Bracket Form 201

14.3 Symplectic or Canonical Transformations 202

14.3a Eigenvalues of Symplectic Matrices 203

14.3b Infinitesimally Symplectic Transformations 204

14.3c The Eigenvalues of Infinitesimally Symplectic Matrices 206

14.3d The Flow Generated by Hamiltonian Vector Fields is a One-Parameter Family of Symplectic Transformations 206

14.4 Transformation of Hamilton's Equations Under Symplectic Transformations 208

14.4a Hamilton's Equations in Complex Coordinates 209

14.5 Completely Integrable Hamiltonian Systems 210

14.6 Dynamics of Completely Integrable Hamiltonian Systems in Action-Angle Coordinates 211

14.6a Resonance and Nonresonance 212

14.6b Diophantine Frequencies 217

14.6c Geometry of the Resonances 220

14.7 Perturbations of Completely Integrable Hamiltonian Systems in Action-Angle Coordinates 221

14.8 Stability of Elliptic Equilibria 222

14.9 Discrete-Time Hamiltonian Dynamical Systems：Iteration of Symplectic Maps 223

14.9a The KAM Theorem and Nekhoroshev's Theorem for Symplectic Maps 223

14.10 Generic Properties of Hamiltonian Dynamical Systems 225

14.11 Exercises 226

15 Gradient Vector Fields 231

15.1 Exercises 232

16 Reversible Dynamical Systems 234

16.1 The Definition of Reversible Dynamical Systems 234

16.2 Examples of Reversible Dynamical Systems 235

16.3 Linearization of Reversible Dynamical Systems 236

16.3a Continuous Time 236

16.3b Discrete Time 238

16.4 Additional Properties of Reversible Dynamical Systems 239

16.5 Exercises 240

17 Asymptotically Autonomous Vector Fields 242

17.1 Exercises 244

18 Center Manifolds 245

18.1 Center Manifolds for Vector Fields 246

18.2 Center Manifolds Depending on Parameters 251

18.3 The Inclusion of Linearly Unstable Directions 256

18.4 Center Manifolds for Maps 257

18.5 Properties of Center Manifolds 263

18.6 Final Remarks on Center Manifolds 265

18.7 Exercises 265

19 Normal Forms 270

19.1 Normal Forms for Vector Fields 270

19.1a Preliminary Preparation of the Equations 270

19.1b Simplification of the Second Order Terms 272

19.1c Simplification of the Third Order Terms 274

19.1d The Normal Form Theorem 275

19.2 Normal Forms for Vector Fields with Parameters 278

19.2a Normal Form for The Poincaré-Andronov-Hopf Bifurcation 279

19.3 Normal Forms for Maps 284

19.3a Normal Form for the Naimark-Sacker Torus Bifurcation 285

19.4 Exercises 288

19.5 The Elphick-Tirapegui-Brachet-Coullet-Iooss Normal Form 290

19.5a An Inner Product on Hk 291

19.5b The Main Theorems 292

19.5c Symmetries of the Normal Form 296

19.5d Examples 298

19.5e The Normal Form of a Vector Field Depending on Parameters 302

19.6 Exercises 304

19.7 Lie Groups,Lie Group Actions,and Symmetries 306

19.7a Examples of Lie Groups 308

19.7b Examples of Lie Group Actions on Vector Spaces 310

19.7c Symmetric Dynamical Systems 312

19.8 Exercises 312

19.9 Normal Form Coefficients 314

19.10 Hamiltonian Normal Forms 316

19.10a General Theory 316

19.10b Normal Forms Near Elliptic Fixed Points：The Semisimple Case 322

19.10c The Birkhoff and Gustavson Normal Forms 333

19.10d The Lyapunov Subcenter Theorem and Moser's Theorem 334

19.10e The KAM and Nekhoroshev Theorem's Near an Elliptic Equilibrium Point 336

19.10f Hamiltonian Normal Forms and Symmetries 338

19.10g Final Remarks 342

19.11 Exercises 342

19.12 Conjugacies and Equivalences of Vector Fields 345

19.12a An Application：The Hartman-Grobman Theorem 350

19.12b An Application：Dynamics Near a Fixed Point-？ita？vili's Theorem 353

19.13 Final Remarks on Normal Forms 353

20 Bifurcation of Fixed Points of Veetor Fields 356

20.1 A Zero Eigenvalue 357

20.1a Examples 358

20.1b What Is A"Bifurcation of a Fixed Point"? 361

20.1c The Saddle-Node Bifurcation 363

20.1d The Transcritical Bifurcation 366

20.1e The Pitchfork Bifurcation 370

20.1f Exercises 373

20.2 A Pure Imaginary Pair of Eigenvalues：The Poincare-Andronov-Hopf Bifurcation 378

20.2a Exercises 386

20.3 Stability of Bifureations Under Perturbations 387

20.4 The Idea of the Codimension of a Bifurcation 392

20.4a The"Big Picture"for Bifurcation Theory 393

20.4b The Approach to Local Bifurcation Theory：Ideas and Results from Singularity Theory 397

20.4c The Codimension of a Local Bifurcation 402

20.4d Construction of Versal Deformations 406

20.4e Exercises 415

20.5 Versal Deformations of Families of Matrices 417

20.5a Versal Deformations of Real Matrices 431

20.5b Exercises 435

20.6 The Double-Zero Eigenvalue：the Takens-Bogdanov Bifurcation 436

20.6a Additional References and Applications for the Takens-Bogdanov Bifurcation 446

20.6b Exercises 446

20.7 A Zero and a Pure Imaginary Pair of Eigenvalues：the Hopf-Steady State Bifurcation 449

20.7a Additional References and Applications for the Hopf-Steady State Bifurcation 477

20.7b Exercises 477

20.8 Versal Deformations of Linear Hamiltonian Systems 482

20.8a Williamson's Theorem 482

20.8b Versal Deformations of Jordan Blocks Corresponding to Repeated Eigenvalues 485

20.8c Versal Deformations of Quadratic Hamiltonians of Codimension≤2 488

20.8d Versal Deformations of Linear.Reversible Dynamical Systems 490

20.8e Exercises 491

20.9 Elementary Hamiltonian Bifurcations 491

20.9a One Degree-of-Freedom Systems 491

20.9b Exercises 494

20.9c Bifureations Near Resonant Elliptic Equilibrium Points 495

20.9d Exercises 497

21 Bifurcations of Fixed Points of Maps 498

21.1 An Eigenvalue of 1 499

21.1a The Saddle-Node Bifurcation 500

21.1b The Transcritical Bifurcation 504

21.1c The Pitchfork Bifurcation 508

21.2 An Eigenvalue of-1：Period Doubling 512

21.2a Example 513

21.2b The Period-Doubling Bifurcation 515

21.3 A Pair of Eigenvalues of Modulus 1：The Naimark-Sacker Bifurcation 517

21.4 The Codimension of Local Bifurcations of Maps 523

21.4a One-Dimensional Maps 524

21.4b Two-Dimensional Maps 524

21.5 Exercises 526

21.6 Maps of the Circle 530

21.6a The Dynamics of a Special Class of Circle Maps-Arnold Tongues 542

21.6b Exercises 550

22 On the Interpretation and Application of Bifurcation Diagrams：A Word of Caution 552

23 The Smale Horseshoe 555

23.1 Definition of the Smale Horseshoe Map 555

23.2 Construction of the Invariant Set 558

23.3 Symbolic Dynamics 566

23.4 The Dynamics on the Invariant Set 570

23.5 Chaos 573

23.6 Final Remarks and Observations 574

24 Symbolic Dynamics 576

24.1 The Structure of the Space of Symbol Sequences 577

24.2 The Shift Map 581

24.3 Exercises 582

25 The Conley-Moser Conditions,or"How to Prove That a Dynamical System is Chaotic" 585

25.1 The Main Theorem 585

25.2 Sector Bundles 602

25.3 Exercises 608

26 Dynamics Near Homoclinic Points of Two-Dimensional Maps 612

26.1 Heteroclinic Cycles 631

26.2 Exercises 632

27 Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields 636

27.1 The Technique of Analysis 637

27.2 Orbits Homoclinic to a Saddle-Point with Purely Real Eigenvalues 640

27.2a Two Orbits Homoclinic to a Fixed Point Having Real Eigenvalues 651

27.2b Observations and Additional References 657

27.3 Orbits Homoclinic to a Saddle-Focus 659

27.3a The Bifurcation Analysis of Glendinning and Sparrow 666

27.3b Double-Pulse Homoclinic Orbits 676

27.3c Observations and General Remarks 676

27.4 Exercises 681

28 Melnikov's Method for Homoclinic Orbits in Two-Dimensional,Time-Periodic Vector Fields 687

28.1 The General Theory 687

28.2 Poincaré Maps and the Geometry of the Melnikov Function 711

28.3 Some Properties ofthe Melnikov Function 713

28.4 Homoclinic Bifurcations 715

28.5 Application to the Damped.Forced Duffing Oscillator 717

28.6 Exercises 720

29 Liapunov Exponents 726

29.1 Liapunov Exponents of a Trajectory 726

29.2 Examples 730

29.3 Numerical Computation of Liapunov Exponents 734

29.4 Exercises 734

30 Chaos and Strange Attractors 736

30.1 Exercises 745

31 Hyperbolic Invariant Sets：A Chaotic Saddle 747

31.1 Hyperbolicity of the Invariant Cantor Set A Constructed in Chapter 25 747

31.1a Stable and Unstable Manifolds of the Hyperbolic Invariant Set 753

31.2 Hyperbolic Invariant Sets in Rn 754

31.2a Sector Bundles for Maps on Rn 757

31.3 A Consequence of Hyperbolicity：The Shadowing Lemma 758

31.3a Applications of the Shadowing Lemma 759

31.4 Exercises 760

32 Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems 762

32.1 Homoclinic Bifurcations 762

32.2 Newhouse Sinks in Dissipative Systems 774

32.3 Islands of Stability in Conservative Systems 776

32.4 Exercises 776

33 Global Bifurcations Arising from Local Codimension—Two Bifurcations 777

33.1 The Double-Zero Eigenvalue 777

33.2 A Zero and a Pure Imaginary Pair of Eigenvalues 782

33.3 Exercises 790

34 Glossary of Frequently Used Terms 793

Bibliography 809

Index 836