Introduction to applied nonlinear dynamical systems and chaos:
"This volume is intended for advanced undergraduate or first-year graduate students as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quanti...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2003
|
| Edition: | 2. ed. |
| Series: | Texts in applied mathematics
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Summary: | "This volume is intended for advanced undergraduate or first-year graduate students as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about the behavior of these systems. He has included the basic core material that is necessary for higher levels of study and research. Thus, people who do not necessarily have an extensive mathematical background, such as students in engineering, physics, chemistry, and biology, will find this text as useful as will students of mathematics."--BOOK JACKET. |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIX, 843 S. graph. Darst. |
| ISBN: | 0387001778 |
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| 490 | 1 | |a Texts in applied mathematics |v 2 | |
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Record in the Search Index
| _version_ | 1804129679397879808 |
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| adam_text | Contents
Series Preface v
Preface to the Second Edition vii
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
xii Contents
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 Poincare Recurrence Theorem 101
7.7 Exercises 101
8 Asymptotic Behavior 104
8.1 The Asymptotic Behavior of Trajectories 104
Contents xiii
8.2 Attracting Sets, Attractors, and Basins of Attraction . . . 107
8.3 The LaSalle Invariance Principle 110
8.4 Attraction in Nonautonomous Systems Ill
8.5 Exercises 114
9 The Poincare Bendixson Theorem 117
9.1 Exercises 121
10 Poincare Maps 122
10.1 Case 1: Poincare Map Near a Periodic Orbit 123
10.2 Case 2: The Poincare Map of a Time Periodic Ordinary
Differential Equation 127
10.2a Periodically Forced Linear Oscillators 128
10.3 Case 3: The Poincare Map Near a Homoclinic Orbit .... 138
10.4 Case 4: Poincare 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: Poincare Map Near a Periodic Orbit: Variation of
the Cross Section 154
11.2 Case 2: The Poincare 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 Form
of Hamilton s Equations 184
13.7d Applications of Variational Methods 186
13.8 The Hamilton Jacobi Equation 187
xiv Contents
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
Contents xv
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.Id The Normal Form Theorem 275
19.2 Normal Forms for Vector Fields with Parameters 278
19.2a Normal Form for The Poincare 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
xvi Contents
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 Sositaisvili s Theorem 353
19.13 Final Remarks on Normal Forms 353
20 Bifurcation of Fixed Points of Vector 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. Id The Transcritical Bifurcation 366
20.le 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 Bifurcations 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
Contents xvii
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
Codimcnsion 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 Bifurcations 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
xviii Contents
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 Poincare Maps and the Geometry of the
Melnikov Function 711
28.3 Some Properties of the Melnikov Function 713
Contents xix
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 Kn 754
31.2a Sector Bundles for Maps on R 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
Contents Series Preface Preface to the Second Edition v vii Introduction 1 1 Equilibrium Solutions, Stability, and Linearized Stability 5 1.1 1.2 1.3 1.4 1.5 1.6 Equilibria of Vector Fields................................................... Stability of Trajectories......................................................... 1.2a Linearization . ............................................................ Maps...................................................................................... 1.3a Definitions of Stability for Maps.............................. 1.3b Stability of Fixed Points of Linear Maps ...... 1.3c Stability of Fixed Points of Maps via the Linear Approximation................................. Some Terminology Associated with Fixed Points............... Application to the Unforced Duffing Oscillator.................. Exercises................................................................................ 2 Liapunov Functions 2.1 Exercises................................................................................ 3 Invariant Manifolds: Linear and Nonlinear Systems 3.1 3.2 3.3 3.4 3.5 Stable, Unstable, and Center Subspaces of Linear, Autonomous Vector Fields................................................... 3.1a Invariance of the Stable, Unstable, and Center Subspaces................................................ 3.1b Some Examples........................................................ Stable, Unstable, and Center Manifolds for Fixed Points of Nonlinear, Autonomous Vector Fields ... 3.2a Invariance of the Graph of a Function: Tangency of the Vector
Field to the Graph............ Maps...................................................................................... Some Examples .................................................................... Existence of Invariant Manifolds: The Main Methods of Proof, and How They Work............................................. 5 7 10 15 15 15 15 16 16 16 20 25 28 29 32 33 37 39 40 41 43
xii Contents 3.5a 3.6 3.7 3.8 Application of These Two Methods to a Concrete Example: Existence of the UnstableManifold .... Time-Dependent Hyperbolic Trajectories and their Stable and Unstable Manifolds........................................................... 3.6a Hyperbolic Trajectories............................................... 3.6b Stable and Unstable Manifolds of Hyperbolic Trajectories ........................................ Invariant Manifolds in a Broader Context............................ Exercises.................................................................................... 45 52 53 56 59 62 4 Periodic Orbits 4.1 Nonexistence of Periodic Orbits for Two-Dimensional, Autonomous Vector Fields..................................................... 4.2 Further Remarks on Periodic Orbits..................................... 4.3 Exercises.................................................................................... 71 5 Vector Fields Possessing an Integral 5.1 Vector Fields on Two-Manifolds Having an Integral .... 5.2 Two Degree-of-Freedom Hamiltonian Systems and Geometry.......................................................................... 5.2a Dynamics on the Energy Surface............................... 5.2b Dynamics on an Individual Torus............................ 5.3 Exercises.................................................................................... 77 77 6 Index Theory 6.1 Exercises............................................................................. 87 89 72 74 76 82 83 85 85 7 Some General Properties of Vector Fields:
Existence, Uniqueness, Differentiability, and Flows 90 7.1 Existence, Uniqueness, Differentiabüity 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 Poincaré Recurrence Theorem ...................... 101 7.7 Exercises....................................................................................... 101 8 Asymptotic Behavior 104 8.1 The Asymptotic Behavior of Trajectories.................................104
Contents 8.2 8.3 8.4 8.5 xiii Attracting Sets, Attractors, and Basins of Attraction . . . 107 The LaSalle Invariance Principle............................................. 110 Attraction in Nonautonomous Systems..............................Ill Exercises.................................................................................... 114 9 The Poincaré-Bendixson Theorem 9.1 117 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 Differential 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 Osciilators 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 Form of Hamilton’s Equations ............................................. 184 13.7d Applications of Variational Methods........................... 186 13.8 The Hamilton-Jacobi Equation................................................ 187
xiv Contents 13.9 13.8a Applications of the Hamilton-Jacobi Equation . . . 192 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
Contents xv 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. Id 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 Я*,............................................... 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
xvi Contents 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.lOf Hamiltonian Normal Forms and Symmetries .... 338 19.10g Final Remarks..................................................................342 19.11 Exercises....................................................................................... 342 19.12 Conjugades and Equivalences of Vector Fields...................... 345 19.12a An Application: The Hartman-Grobman Theorem...........................................................................350 19.12b An Application: Dynamics Near a Fixed Point-Šošitaišvili’s Theorem .........................................353 19.13 Final Remarks on Normal Forms............................................... 353 20 Bifurcation of Fixed Points of Vector 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
Transcriticai Bifurcation........................................366 20. le The Pitchfork Bifurcation.............................................. 370 20. If Exercises.......................................................................... 373 20.2 A Pure Imaginary Pair of Eigenvalues: The Poincare-Andronov-Hopf Bifurcation............................... 378 20.2a Exercises.......................................................................... 386 20.3 Stability of Bifurcations 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
Contents xvii 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 Bifurcations 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 Transcriticai 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
xviii Contents 23 The 23.1 23.2 23.3 23.4 23.5 23.6 Smale Horseshoe 555 Definition of the Smale Horseshoe Map...................................555 Construction of the Invariant Set............................................... 558 Symbolic Dynamics.....................................................................566 The Dynamics on the Invariant Set .........................................570 Chaos............................................................................................. 573 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 of the Melnikov Function .............................713
Contents xix 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 Λ Constructed in Chapter 25 ................................................................................. 747 31.1a Stable and Unstable Manifolds of the Hyperbolic Invariant Set.....................................................................753 31.2 Hyperbolic Invariant Sets in M” ............................................... 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
|
| any_adam_object | 1 |
| author | Wiggins, Stephen ca. 20./21. Jh |
| author_GND | (DE-588)1247764664 |
| author_facet | Wiggins, Stephen ca. 20./21. Jh |
| author_role | aut |
| author_sort | Wiggins, Stephen ca. 20./21. Jh |
| author_variant | s w sw |
| building | Verbundindex |
| bvnumber | BV015022675 |
| callnumber-first | Q - Science |
| callnumber-label | QA614 |
| callnumber-raw | QA614.8.W544 2003 |
| callnumber-search | QA614.8.W544 2003 |
| callnumber-sort | QA 3614.8 W544 42003 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | CC 6300 SK 350 SK 520 SK 540 SK 810 UG 3900 |
| classification_tum | MAT 587f MAT 344f |
| ctrlnum | (OCoLC)51093130 (DE-599)BVBBV015022675 |
| dewey-full | 003/.85 003/.8521 |
| dewey-hundreds | 000 - Computer science, information, general works |
| dewey-ones | 003 - Systems |
| dewey-raw | 003/.85 003/.85 21 |
| dewey-search | 003/.85 003/.85 21 |
| dewey-sort | 13 285 |
| dewey-tens | 000 - Computer science, information, general works |
| discipline | Physik Informatik Mathematik Philosophie |
| edition | 2. ed. |
| format | Book |
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| genre | 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content |
| genre_facet | Aufgabensammlung |
| id | DE-604.BV015022675 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:08:58Z |
| institution | BVB |
| isbn | 0387001778 |
| language | English |
| lccn | 2002042742 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010091542 |
| oclc_num | 51093130 |
| open_access_boolean | |
| owner | DE-703 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-473 DE-BY-UBG DE-521 DE-634 DE-83 DE-11 DE-188 DE-859 DE-20 DE-739 DE-29T |
| owner_facet | DE-703 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-473 DE-BY-UBG DE-521 DE-634 DE-83 DE-11 DE-188 DE-859 DE-20 DE-739 DE-29T |
| physical | XIX, 843 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Texts in applied mathematics |
| series2 | Texts in applied mathematics |
| spelling | Wiggins, Stephen ca. 20./21. Jh. Verfasser (DE-588)1247764664 aut Introduction to applied nonlinear dynamical systems and chaos Stephen Wiggins 2. ed. New York [u.a.] Springer 2003 XIX, 843 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics 2 Includes bibliographical references and index "This volume is intended for advanced undergraduate or first-year graduate students as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about the behavior of these systems. He has included the basic core material that is necessary for higher levels of study and research. Thus, people who do not necessarily have an extensive mathematical background, such as students in engineering, physics, chemistry, and biology, will find this text as useful as will students of mathematics."--BOOK JACKET. Differentiable dynamical systems Nonlinear theories Chaotic behavior in systems Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Nichtlineare Theorie (DE-588)4251279-7 gnd rswk-swf 1\p (DE-588)4143389-0 Aufgabensammlung gnd-content Chaotisches System (DE-588)4316104-2 s Nichtlineares dynamisches System (DE-588)4126142-2 s DE-604 Chaostheorie (DE-588)4009754-7 s 2\p DE-604 Chaos (DE-588)4191419-3 s 3\p DE-604 Nichtlineare Theorie (DE-588)4251279-7 s 4\p DE-604 Differenzierbares dynamisches System (DE-588)4137931-7 s 5\p DE-604 Texts in applied mathematics 2 (DE-604)BV002476038 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091542&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091542&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Wiggins, Stephen ca. 20./21. Jh Introduction to applied nonlinear dynamical systems and chaos Texts in applied mathematics Differentiable dynamical systems Nonlinear theories Chaotic behavior in systems Nichtlineares dynamisches System (DE-588)4126142-2 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Chaos (DE-588)4191419-3 gnd Chaostheorie (DE-588)4009754-7 gnd Chaotisches System (DE-588)4316104-2 gnd Nichtlineare Theorie (DE-588)4251279-7 gnd |
| subject_GND | (DE-588)4126142-2 (DE-588)4137931-7 (DE-588)4191419-3 (DE-588)4009754-7 (DE-588)4316104-2 (DE-588)4251279-7 (DE-588)4143389-0 |
| title | Introduction to applied nonlinear dynamical systems and chaos |
| title_auth | Introduction to applied nonlinear dynamical systems and chaos |
| title_exact_search | Introduction to applied nonlinear dynamical systems and chaos |
| title_full | Introduction to applied nonlinear dynamical systems and chaos Stephen Wiggins |
| title_fullStr | Introduction to applied nonlinear dynamical systems and chaos Stephen Wiggins |
| title_full_unstemmed | Introduction to applied nonlinear dynamical systems and chaos Stephen Wiggins |
| title_short | Introduction to applied nonlinear dynamical systems and chaos |
| title_sort | introduction to applied nonlinear dynamical systems and chaos |
| topic | Differentiable dynamical systems Nonlinear theories Chaotic behavior in systems Nichtlineares dynamisches System (DE-588)4126142-2 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd Chaos (DE-588)4191419-3 gnd Chaostheorie (DE-588)4009754-7 gnd Chaotisches System (DE-588)4316104-2 gnd Nichtlineare Theorie (DE-588)4251279-7 gnd |
| topic_facet | Differentiable dynamical systems Nonlinear theories Chaotic behavior in systems Nichtlineares dynamisches System Differenzierbares dynamisches System Chaos Chaostheorie Chaotisches System Nichtlineare Theorie Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091542&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010091542&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT wigginsstephen introductiontoappliednonlineardynamicalsystemsandchaos |
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