非线性动力系统和混沌应用导论 第2版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

非线性动力系统和混沌应用导论 第2版PDF格式电子书版下载

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Introduction1

1 Equilibrium Solutions,Stability,and Linearized Stability5

1.1 Equilibria of Vector Fields5

1.2 Stability of Trajectories7

1.2a Linearization10

1.3 Maps15

1.3a Definitions of Stability for Maps15

1.3b Stability of Fixed Points of Linear Maps15

1.3c Stability of Fixed Points of Maps via the Linear Approximation15

1.4 Some Terminology Associated with Fixed Points16

1.5 Application to the Unforced Duffing Oscillator16

1.6 Exercises16

2 Liapunov Functions20

2.1 Exercises25

3 Invariant Manifolds：Linear and Nonlinear Systems28

3.1 Stable,Unstable,and Center Subspaces of Linear,Autonomous Vector Fields29

3.1a Invariance of the Stable,Unstable,and Center Subspaces32

3.1b Some Examples33

3.2 Stable,Unstable,and Center Manifolds for Fixed Points of Nonlinear,Autonomous Vector Fields37

3.2a Invariance of the Graph of a Function：Tangency of the Vector Field to the Graph39

3.3 Maps40

3.4 Some Examples41

3.5 Existence of Invariant Manifolds：The Main Methods of Proof,and How They Work43

3.5a Application of These Two Methods to a Concrete Example：Existence of the Unstable Manifold45

3.6 Time-Dependent Hyperbolic Trajectories and their Stable and Unstable Manifolds52

3.6a Hyperbolic Trajectories53

3.6b Stable and Unstable Manifolds of Hyperbolic Trajectories56

3.7 Invariant Manifolds in a Broader Context59

3.8 Exercises62

4 Periodic Orbits71

4.1 Nonexistence of Periodic Orbits for Two-Dimensional,Autonomous Vector Fields72

4.2 Further Remarks on Periodic Orbits74

4.3 Exercises76

5 Vector Fields Possessing an Integral77

5.1 Vector Fields on Two-Manifolds Having an Integral77

5.2 Two Degree-of-Freedom Hamiltonian Systems and Geometry82

5.2a Dynamics on the Energy Surface83

5.2b Dynamics on an Individual Torus85

5.3 Exercises85

6 Index Theory87

6.1 Exercises89

7 Some General Properties of Vector Fields：Existence,Uniqueness,Differentiability,and Flows90

7.1 Existence,Uniqueness,Differentiability with Respect to Initial Conditions90

7.2 Continuation of Solutions91

7.3 Differentiability with Respect to Parameters91

7.4 Autonomous Vector Fields92

7.5 Nonautonomous Vector Fields94

7.5a The Skew-Product Flow Approach95

7.5b The Cocycle Approach97

7.5c Dynamics Generated by a Bi-Infinite Sequence of Maps97

7.6 Liouville's Theorem99

7.6a Volume Preserving Vector Fields and the PoincaréRecurrence Theorem101

7.7 Exercises101

8 Asymptotic Behavior104

8.1 The Asymptotic Behavior of Trajectories104

8.2 Attracting Sets,Attractors,and Basins of Attraction107

8.3 The LaSalle Invariance Principle110

8.4 Attraction in Nonautonomous Systems111

8.5 Exercises114

9 The Poincaré-Bendixson Theorem117

9.1 Exercises121

10 Poincaré Maps122

10.1 Case 1：Poincaré Map Near a Periodic Orbit123

10.2 Case 2：The Poincaré Map of a Time-Periodic Ordinary Difierential Equation127

10.2a Periodically Forced Linear Oscillators128

10.3 Case 3：The Poincaré Map Near a Homoclinic Orbit138

10.4 Case 4：Poincaré Map Associated with a Two Degree-of-Freedom Hamiltonian System144

10.4a The Study of Coupled Oscillators via Circle Maps146

10.5 Exercises149

11 Conjugacies of Maps,and Varying the Cross-Section151

11.1 Case 1：Poincaré Map Near a Periodic Orbit：Variation of the Cross-Section154

11.2 Case 2：The Poincaré Map of a Time-Periodic Ordinary Differential Equation：Variation of the Cross-Section155

12 Structural Stability,Genericity,and Transversality157

12.1 Definitions of Structural Stability and Genericity161

12.2 Transversality165

12.3 Exercises167

13 Lagrange's Equations169

13.1 Generalized Coordinates170

13.2 Derivation of Lagrange's Equations172

13.2a The Kinetic Energy175

13.3 The Energy Integral176

13.4 Momentum Integrals177

13.5 Hamilton's Equations177

13.6 Cyclic Coordinates,Routh's Equations,and Reduction of the Number of Equations178

13.7 Variational Methods180

13.7a The Principle of Least Action180

13.7b The Action Principle in Phase Space182

13.7c Transformations that Preserve the Fcrm of Hamilton's Equations184

13.7d Applications of Variational Methods186

13.8 The Hamilton-Jacobi Equation187

13.8a Applications of the Hamilton-Jacobi Equation192

13.9 Exercises192

14 Hamiltonian Vector Fields197

14.1 Symplectic Forms199

14.1a The Relationship Between Hamilton's Equations and the Symplectic Form199

14.2 Poisson Brackets200

14.2a Hamilton's Equations in Poisson Bracket Form201

14.3 Symplectic or Canonical Transformations202

14.3a Eigenvalues of Symplectic Matrices203

14.3b Infinitesimally Symplectic Transformations204

14.3c The Eigenvalues of Infinitesimally Symplectic Matrices206

14.3d The Flow Generated by Hamiltonian Vector Fields is a One-Parameter Family of Symplectic Transformations206

14.4 Transformation of Hamilton's Equations Under Symplectic Transformations208

14.4a Hamilton's Equations in Complex Coordinates209

14.5 Completely Integrable Hamiltonian Systems210

14.6 Dynamics of Completely Integrable Hamiltonian Systems in Action-Angle Coordinates211

14.6a Resonance and Nonresonance212

14.6b Diophantine Frequencies217

14.6c Geometry of the Resonances220

14.7 Perturbations of Completely Integrable Hamiltonian Systems in Action-Angle Coordinates221

14.8 Stability of Elliptic Equilibria222

14.9 Discrete-Time Hamiltonian Dynamical Systems：Iteration of Symplectic Maps223

14.9a The KAM Theorem and Nekhoroshev's Theorem for Symplectic Maps223

14.10 Generic Properties of Hamiltonian Dynamical Systems225

14.11 Exercises226

15 Gradient Vector Fields231

15.1 Exercises232

16 Reversible Dynamical Systems234

16.1 The Definition of Reversible Dynamical Systems234

16.2 Examples of Reversible Dynamical Systems235

16.3 Linearization of Reversible Dynamical Systems236

16.3a Continuous Time236

16.3b Discrete Time238

16.4 Additional Properties of Reversible Dynamical Systems239

16.5 Exercises240

17 Asymptotically Autonomous Vector Fields242

17.1 Exercises244

18 Center Manifolds245

18.1 Center Manifolds for Vector Fields246

18.2 Center Manifolds Depending on Parameters251

18.3 The Inclusion of Linearly Unstable Directions256

18.4 Center Manifolds for Maps257

18.5 Properties of Center Manifolds263

18.6 Final Remarks on Center Manifolds265

18.7 Exercises265

19 Normal Forms270

19.1 Normal Forms for Vector Fields270

19.1a Preliminary Preparation of the Equations270

19.1b Simplification of the Second Order Terms272

19.1c Simplification of the Third Order Terms274

19.1d The Normal Form Theorem275

19.2 Normal Forms for Vector Fields with Parameters278

19.2a Normal Form for The Poincaré-Andronov-Hopf Bifurcation279

19.3 Normal Forms for Maps284

19.3a Normal Form for the Naimark-Sacker Torus Bifurcation285

19.4 Exercises288

19.5 The Elphick-Tirapegui-Brachet-Coullet-Iooss Normal Form290

19.5a An Inner Product on Hk291

19.5b The Main Theorems292

19.5c Symmetries of the Normal Form296

19.5d Examples298

19.5e The Normal Form of a Vector Field Depending on Parameters302

19.6 Exercises304

19.7 Lie Groups,Lie Group Actions,and Symmetries306

19.7a Examples of Lie Groups308

19.7b Examples of Lie Group Actions on Vector Spaces310

19.7c Symmetric Dynamical Systems312

19.8 Exercises312

19.9 Normal Form Coefficients314

19.10 Hamiltonian Normal Forms316

19.10a General Theory316

19.10b Normal Forms Near Elliptic Fixed Points：The Semisimple Case322

19.10c The Birkhoff and Gustavson Normal Forms333

19.10d The Lyapunov Subcenter Theorem and Moser's Theorem334

19.10e The KAM and Nekhoroshev Theorem's Near an Elliptic Equilibrium Point336

19.10f Hamiltonian Normal Forms and Symmetries338

19.10g Final Remarks342

19.11 Exercises342

19.12 Conjugacies and Equivalences of Vector Fields345

19.12a An Application：The Hartman-Grobman Theorem350

19.12b An Application：Dynamics Near a Fixed Point-？ita？vili's Theorem353

19.13 Final Remarks on Normal Forms353

20 Bifurcation of Fixed Points of Veetor Fields356

20.1 A Zero Eigenvalue357

20.1a Examples358

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

20.1c The Saddle-Node Bifurcation363

20.1d The Transcritical Bifurcation366

20.1e The Pitchfork Bifurcation370

20.1f Exercises373

20.2 A Pure Imaginary Pair of Eigenvalues：The Poincare-Andronov-Hopf Bifurcation378

20.2a Exercises386

20.3 Stability of Bifureations Under Perturbations387

20.4 The Idea of the Codimension of a Bifurcation392

20.4a The"Big Picture"for Bifurcation Theory393

20.4b The Approach to Local Bifurcation Theory：Ideas and Results from Singularity Theory397

20.4c The Codimension of a Local Bifurcation402

20.4d Construction of Versal Deformations406

20.4e Exercises415

20.5 Versal Deformations of Families of Matrices417

20.5a Versal Deformations of Real Matrices431

20.5b Exercises435

20.6 The Double-Zero Eigenvalue：the Takens-Bogdanov Bifurcation436

20.6a Additional References and Applications for the Takens-Bogdanov Bifurcation446

20.6b Exercises446

20.7 A Zero and a Pure Imaginary Pair of Eigenvalues：the Hopf-Steady State Bifurcation449

20.7a Additional References and Applications for the Hopf-Steady State Bifurcation477

20.7b Exercises477

20.8 Versal Deformations of Linear Hamiltonian Systems482

20.8a Williamson's Theorem482

20.8b Versal Deformations of Jordan Blocks Corresponding to Repeated Eigenvalues485

20.8c Versal Deformations of Quadratic Hamiltonians of Codimension≤2488

20.8d Versal Deformations of Linear.Reversible Dynamical Systems490

20.8e Exercises491

20.9 Elementary Hamiltonian Bifurcations491

20.9a One Degree-of-Freedom Systems491

20.9b Exercises494

20.9c Bifureations Near Resonant Elliptic Equilibrium Points495

20.9d Exercises497

21 Bifurcations of Fixed Points of Maps498

21.1 An Eigenvalue of 1499

21.1a The Saddle-Node Bifurcation500

21.1b The Transcritical Bifurcation504

21.1c The Pitchfork Bifurcation508

21.2 An Eigenvalue of-1：Period Doubling512

21.2a Example513

21.2b The Period-Doubling Bifurcation515

21.3 A Pair of Eigenvalues of Modulus 1：The Naimark-Sacker Bifurcation517

21.4 The Codimension of Local Bifurcations of Maps523

21.4a One-Dimensional Maps524

21.4b Two-Dimensional Maps524

21.5 Exercises526

21.6 Maps of the Circle530

21.6a The Dynamics of a Special Class of Circle Maps-Arnold Tongues542

21.6b Exercises550

22 On the Interpretation and Application of Bifurcation Diagrams：A Word of Caution552

23 The Smale Horseshoe555

23.1 Definition of the Smale Horseshoe Map555

23.2 Construction of the Invariant Set558

23.3 Symbolic Dynamics566

23.4 The Dynamics on the Invariant Set570

23.5 Chaos573

23.6 Final Remarks and Observations574

24 Symbolic Dynamics576

24.1 The Structure of the Space of Symbol Sequences577

24.2 The Shift Map581

24.3 Exercises582

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

25.1 The Main Theorem585

25.2 Sector Bundles602

25.3 Exercises608

26 Dynamics Near Homoclinic Points of Two-Dimensional Maps612

26.1 Heteroclinic Cycles631

26.2 Exercises632

27 Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields636

27.1 The Technique of Analysis637

27.2 Orbits Homoclinic to a Saddle-Point with Purely Real Eigenvalues640

27.2a Two Orbits Homoclinic to a Fixed Point Having Real Eigenvalues651

27.2b Observations and Additional References657

27.3 Orbits Homoclinic to a Saddle-Focus659

27.3a The Bifurcation Analysis of Glendinning and Sparrow666

27.3b Double-Pulse Homoclinic Orbits676

27.3c Observations and General Remarks676

27.4 Exercises681

28 Melnikov's Method for Homoclinic Orbits in Two-Dimensional,Time-Periodic Vector Fields687

28.1 The General Theory687

28.2 Poincaré Maps and the Geometry of the Melnikov Function711

28.3 Some Properties ofthe Melnikov Function713

28.4 Homoclinic Bifurcations715

28.5 Application to the Damped.Forced Duffing Oscillator717

28.6 Exercises720

29 Liapunov Exponents726

29.1 Liapunov Exponents of a Trajectory726

29.2 Examples730

29.3 Numerical Computation of Liapunov Exponents734

29.4 Exercises734

30 Chaos and Strange Attractors736

30.1 Exercises745

31 Hyperbolic Invariant Sets：A Chaotic Saddle747

31.1 Hyperbolicity of the Invariant Cantor Set A Constructed in Chapter 25747

31.1a Stable and Unstable Manifolds of the Hyperbolic Invariant Set753

31.2 Hyperbolic Invariant Sets in Rn754

31.2a Sector Bundles for Maps on Rn757

31.3 A Consequence of Hyperbolicity：The Shadowing Lemma758

31.3a Applications of the Shadowing Lemma759

31.4 Exercises760

32 Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems762

32.1 Homoclinic Bifurcations762

32.2 Newhouse Sinks in Dissipative Systems774

32.3 Islands of Stability in Conservative Systems776

32.4 Exercises776

33 Global Bifurcations Arising from Local Codimension—Two Bifurcations777

33.1 The Double-Zero Eigenvalue777

33.2 A Zero and a Pure Imaginary Pair of Eigenvalues782

33.3 Exercises790

34 Glossary of Frequently Used Terms793

Bibliography809

Index836