The Topology of Chaos: Alice in Stretch and Squeezeland Book

	Preface
1	Introduction	1
1.1	Laser with Modulated Losses	2
1.2	Objectives of a New Analysis Procedure	10
1.3	Preview of Results	11
1.4	Organization of This Work	12
2	Discrete Dynamical Systems: Maps	17
2.1	Introduction	17
2.2	Logistic Map	19
2.3	Bifurcation Diagrams	21
2.4	Elementary Bifurcations in the Logistic Map	23
2.5	Map Conjugacy	30
2.6	Fully Developed Chaos in the Logistic Map	32
2.7	One-Dimensional Symbolic Dynamics	40
2.8	Shift Dynamical Systems, Markov Partitions, and Entropy	57
2.9	Fingerprints of Periodic Orbits and Orbit Forcing	67
2.10	Two-Dimensional Dynamics: Smale's Horseshoe	74
2.11	Henon Map	82
2.12	Circle Maps	90
2.13	Summary	95
3	Continuous Dynamical Systems: Flows	97
3.1	Definition of Dynamical Systems	97
3.2	Existence and Uniqueness Theorem	98
3.3	Examples of Dynamical Systems	99
3.4	Change of Variables	112
3.5	Fixed Points	116
3.6	Periodic Orbits	121
3.7	Flows near Nonsingular Points	124
3.8	Volume Expansion and Contraction	125
3.9	Stretching and Squeezing	126
3.10	The Fundamental Idea	127
3.11	Summary	128
4	Topological Invariants	131
4.1	Stretching and Squeezing Mechanisms	132
4.2	Linking Numbers	136
4.3	Relative Rotation Rates	149
4.4	Relation between Linking Numbers and Relative Rotation Rates	159
4.5	Additional Uses of Topological Invariants	160
4.6	Summary	164
5	Branched Manifolds	165
5.1	Closed Loops	166
5.2	What Has This Got to Do with Dynamical Systems?	169
5.3	General Properties of Branched Manifolds	169
5.4	Birman-Williams Theorem	171
5.5	Relaxation of Restrictions	175
5.6	Examples of Branched Manifolds	176
5.7	Uniqueness and Nonuniqueness	186
5.8	Standard Form	190
5.9	Topological Invariants	193
5.10	Additional Properties	199
5.11	Subtemplates	207
5.12	Summary	215
6	Topological Analysis Program	217
6.1	Brief Summary of the Topological Analysis Program	217
6.2	Overview of the Topological Analysis Program	218
6.3	Data	225
6.4	Embeddings	233
6.5	Periodic Orbits	246
6.6	Computation of Topological Invariants	251
6.7	Identify Template	252
6.8	Validate Template	253
6.9	Model Dynamics	254
6.10	Validate Model	257
6.11	Summary	259
7	Folding Mechanisms: A[subscript 2]	261
7.1	Belousov-Zhabotinskii Chemical Reaction	262
7.2	Laser with Saturable Absorber	275
7.3	Stringed Instrument	279
7.4	Lasers with Low-Intensity Signals	284
7.5	The Lasers in Lille	288
7.6	Neuron with Subthreshold Oscillations	315
7.7	Summary	321
8	Tearing Mechanisms: A[subscript 3]	323
8.1	Lorenz Equations	324
8.2	Optically Pumped Molecular Laser	329
8.3	Fluid Experiments	338
8.4	Why A[subscript 3]?	341
8.5	Summary	341
9	Unfoldings	343
9.1	Catastrophe Theory as a Model	344
9.2	Unfolding of Branched Manifolds: Branched Manifolds as Germs	348
9.3	Unfolding within Branched Manifolds: Unfolding of the Horseshoe	351
9.4	Missing Orbits	362
9.5	Routes to Chaos	363
9.6	Summary	365
10	Symmetry	367
10.1	Information Loss and Gain	368
10.2	Cover and Image Relations	369
10.3	Rotation Symmetry 1: Images	370
10.4	Rotation Symmetry 2: Covers	376
10.5	Peeling: A New Global Bifurcation	380
10.6	Inversion Symmetry: Driven Oscillators	383
10.7	Duffing Oscillator	386
10.8	van der Pol Oscillator	389
10.9	Summary	395
11	Flows in Higher Dimensions	397
11.1	Review of Classification Theory in R[superscript 3]	397
11.2	General Setup	399
11.3	Flows in R[superscript 4]	402
11.4	Cusp Bifurcation Diagrams	406
11.5	Nonlocal Singularities	411
11.6	Global Boundary Conditions	414
11.7	Summary	418
12	Program for Dynamical Systems Theory	421
12.1	Reduction of Dimension	422
12.2	Equivalence	425
12.3	Structure Theory	426
12.4	Germs	427
12.5	Unfolding	428
12.6	Paths	430
12.7	Rank	431
12.8	Complex Extensions	432
12.9	Coxeter-Dynkin Diagrams	433
12.10	Real Forms	434
12.11	Local vs. Global Classification	436
12.12	Cover-Image Relations	437
12.13	Symmetry Breaking and Restoration	437
12.14	Summary	439
App. A	Determining Templates from Topological Invariants	441
	References	469
	Topic Index	483