The User's Approach to Topological Methods in 3D Dynamical Systems

Contents......Page 12
Preface......Page 8
Acknowledgments......Page 10
1.1 The Experimental Method Archetype......Page 16
1.2 Deterministic Chaos......Page 17
1.3 Model Validation......Page 18
1.4 The Language of Nonlinear Dynamics......Page 20
1.5 Stereotype Examples of Chaotic Dynamics......Page 23
1.5.1 Smale’s horseshoe......Page 24
1.5.2 The Lorenz equations......Page 26
1.6 Seeking a Way Out / Gathering the Loose Ends......Page 27
1.6.1 A word of warning......Page 28
2.1 Examples of Dynamical Systems in R2 × S1......Page 30
2.1.2 Laser with modulated losses......Page 31
2.2 Homotopies and Topological Properties......Page 32
2.3 Periodic Orbits as Knots......Page 33
2.4 Periodic Orbits as Braids......Page 38
2.4.2 The braid group I......Page 39
2.5 Coloured Braids, Linking Numbers and Relative Rotation Rates......Page 40
2.5.1 Matrix representation of braids......Page 41
2.5.2 Relative rotation rates......Page 42
2.6 The Knot Holder......Page 44
2.6.1 Applications......Page 45
2.7 Appendix: The Horseshoe Template and Orbit Classification......Page 47
3. Braids as Indicators of Phase-space Dynamics......Page 52
3.1 Topological Entropy......Page 53
3.2 Thurston’s Theorem......Page 54
3.2.2 The theorem......Page 55
3.2.3 Orbits That Imply Positive Topological Entropy......Page 57
3.3 Highly Dissipative Systems......Page 58
4. Braids and the Poincaré Section......Page 60
4.1 Braids on the Poincaré Section......Page 61
4.1.1 “Braidless” braids......Page 63
4.2 The Fat Representative of a Pseudo-Anosov Map......Page 64
4.2.1 An algorithm......Page 65
4.3 Trees, Topological Entropy and Orbit Forcing......Page 71
4.3.2 Orbit pruning......Page 72
4.4.2 Second example......Page 75
4.4.3 Third example......Page 76
4.4.4 Fourth (last) example......Page 77
5.1 Introduction: Naive Measurements......Page 80
5.2.1 Filtering and interpolation......Page 81
5.2.2 Close returns......Page 82
5.2.2.1 Guided example......Page 83
5.2.3 Imbedding(s)......Page 86
5.2.4 Imbeddings and phase-space reconstruction......Page 88
5.3 Embedology......Page 90
5.3.2 Performance tests......Page 92
5.3.2.2 False neighbours......Page 93
5.3.2.4 Fractal dimension......Page 94
5.3.2.5 Surrogate data......Page 95
5.4 Reconstruction of the Poincaré Map......Page 96
5.4.1 Sampling the Poincaré map......Page 97
5.4.2 Finding the Markov partition on the Poincaré section......Page 99
5.5 Occam’s Razor......Page 100
5.6 Final Remarks......Page 101
6.1.1 Epistemological ruminations......Page 102
6.2 Templates, Braids and Braid Words......Page 104
6.3 Knots vs Braids: Freedom of Choice of Poincaré Section......Page 105
6.4 Topologically Inequivalent Imbeddings......Page 107
6.5.1 Imbeddings and reconstruction of the dynamics......Page 109
6.5.1.1 A theorem on periodically forced oscillators......Page 110
6.5.2 Imbedding as a coordinate transformation......Page 112
6.5.3 Coordinate transformations and imbeddings from another point of view......Page 113
6.5.4 Symmetries......Page 115
6.6 Higher Dimensions: What is Possible?......Page 116
6.6.1 Local torsion......Page 118
6.6.2 Topological entropy......Page 119
6.6.3 Homology groups......Page 120
7. The User’s Chapter......Page 122
7.1 Laser Physics......Page 123
7.2.1 Biological application......Page 125
7.2.3 Plasma physics......Page 126
8. After Thoughts......Page 128
Bibliography......Page 132
Index......Page 140