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