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E-raamat: Topology of Chaos: Alice in Stretch and Squeezeland

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(Drexel University, US), (University of Lille, France)
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  • Ilmumisaeg: 19-Sep-2012
  • Kirjastus: Blackwell Verlag GmbH
  • Keel: eng
  • ISBN-13: 9783527639427
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Topology of Chaos: Alice in Stretch and SqueezelandSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 19-Sep-2012
  • Kirjastus: Blackwell Verlag GmbH
  • Keel: eng
  • ISBN-13: 9783527639427
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A highly valued resource for those who wish to move from the introductory and preliminary understandings and the measurement of chaotic behavior to a more sophisticated and precise understanding of chaotic systems. The authors provide a deep understanding of the structure of strange attractors, how they are classified, and how the information required to identify and classify a strange attractor can be extracted from experimental data. In its first edition, the Topology of Chaos has been a valuable resource for physicist and mathematicians interested in the topological analysis of dynamical systems. Since its publication in 2002, important theoretical and experimental advances have put the topological analysis program on a firmer basis. This second edition includes relevant results and connects the material to other recent developments. Following significant improvements will be included: * A gentler introduction to the topological analysis of chaotic systems for the non expert which introduces the problems and questions that one commonly encounters when observing a chaotic dynamics and which are well addressed by a topological approach: existence of unstable periodic orbits, bifurcation sequences, multistability etc. * A new chapter is devoted to bounding tori which are essential for achieving generality as well as for understanding the influence of boundary conditions. * The new edition also reflects the progress which had been made towards extending topological analysis to higher-dimensional systems by proposing a new formalism where evolving triangulations replace braids. * There has also been much progress in the understanding of what is a good representation of a chaotic system, and therefore a new chapter is devoted to embeddings. * The chapter on topological analysis program will be expanded to cover traditional measures of chaos. This will help to connect those readers who are familiar with those measures and tests to the more sophisticated methodologies discussed in detail in this book. * The addition of the Appendix with both frequently asked and open questions with answers gathers the most essential points readers should keep in mind and guides to corresponding sections in the book. This will be of great help to those who want to selectively dive into the book and its treatments rather than reading it cover to cover.

What makes this book special is its attempt to classify real physical systems (e.g. lasers) using topological techniques applied to real date (e.g. time series). Hence it has become the experimenter s guidebook to reliable and sophisticated studies of experimental data for comparison with candidate relevant theoretical models, inevitable to physicists, mathematicians, and engineers studying low-dimensional chaotic systems.

Arvustused

On the first edition

"A short review can only hint at the wealth of ideas here...highly recommended." (Choice, Vol. 40, No. 7, March 2003)

"In this third book Gilmore and Lefranc step one more rung up the ladder of dynamical complexity..." (American Journal of Physics, Vol. 71, No. 5, May 2003)

"This authoritative monograph advances innovative methods for the analysis of chaotic systems." (Journal of Mathematical Psychology, Vol. 47, 2003)

Preface to Second Edition xvii
Preface to the First Edition xix
1 Introduction
1(18)
1.1 Brief Review of Useful Concepts
2(2)
1.2 Laser with Modulated Losses
4(7)
1.3 Objectives of a New Analysis Procedure
11(1)
1.4 Preview of Results
12(2)
1.5 Organization of This Work
14(5)
2 Discrete Dynamical Systems: Maps
19(86)
2.1 Introduction
19(1)
2.2 Logistic Map
20(2)
2.3 Bifurcation Diagrams
22(3)
2.4 Elementary Bifurcations in the Logistic Map
25(7)
2.4.1 Saddle-Node Bifurcation
25(4)
2.4.2 Period-Doubling Bifurcation
29(3)
2.5 Map Conjugacy
32(2)
2.5.1 Changes of Coordinates
32(1)
2.5.2 Invariants of Conjugacy
33(1)
2.6 Fully Developed Chaos in the Logistic Map
34(8)
2.6.1 Iterates of the Tent Map
35(1)
2.6.2 Lyapunov Exponents
36(1)
2.6.3 Sensitivity to Initial Conditions and Mixing
37(1)
2.6.4 Chaos and Density of (Unstable) Periodic Orbits
38(1)
2.6.4.1 Number of Periodic Orbits of the Tent Map
38(1)
2.6.4.2 Expansiveness Implies Infinitely Many Periodic Orbits
39(1)
2.6.5 Symbolic Coding of Trajectories: First Approach
40(2)
2.7 One-Dimensional Symbolic Dynamics
42(17)
2.7.1 Partitions
42(2)
2.7.2 Symbolic Dynamics of Expansive Maps
44(4)
2.7.3 Grammar of Chaos: First Approach
48(1)
2.7.3.1 Interval Arithmetics and Invariant Interval
48(1)
2.7.3.2 Existence of Forbidden Sequences
49(2)
2.7.4 Kneading Theory
51(1)
2.7.4.1 Ordering of Itineraries
52(2)
2.7.4.2 Admissible Sequences
54(1)
2.7.5 Bifurcation Diagram of the Logistic Map Revisited
55(1)
2.7.5.1 Saddle-Node Bifurcations
55(1)
2.7.5.2 Period-Doubling Bifurcations
56(1)
2.7.5.3 Universal Sequence
57(1)
2.7.5.4 Self-Similar Structure of the Bifurcation Diagram
58(1)
2.8 Shift Dynamical Systems, Markov Partitions, and Entropy
59(11)
2.8.1 Shifts of Finite Type and Topological Markov Chains
59(2)
2.8.2 Periodic Orbits and Topological Entropy of a Markov Chain
61(2)
2.8.3 Markov Partitions
63(2)
2.8.4 Approximation by Markov Chains
65(1)
2.8.5 Zeta Function
65(1)
2.8.6 Dealing with Grammars
66(1)
2.8.6.1 Simple Grammars
67(2)
2.8.6.2 Complicated Grammars
69(1)
2.9 Fingerprints of Periodic Orbits and Orbit Forcing
70(7)
2.9.1 Permutation of Periodic Points as a Topological Invariant
70(2)
2.9.2 Topological Entropy of a Periodic Orbit
72(2)
2.9.3 Period 3 Implies Chaos and Sarkovskii's Theorem
74(1)
2.9.4 Period 3 Does Not Always Imply Chaos: Role of Phase-Space Topology
75(1)
2.9.5 Permutations and Orbit Forcing
75(2)
2.10 Two-Dimensional Dynamics: Smale's Horseshoe
77(8)
2.10.1 Horseshoe Map
77(1)
2.10.2 Symbolic Dynamics of the Invariant Set
78(3)
2.10.3 Dynamical Properties
81(1)
2.10.4 Variations on the Horseshoe Map: Baker Maps
82(3)
2.11 Henon Map
85(11)
2.11.1 A Once-Folding Map
85(2)
2.11.2 Symbolic Dynamics of the Henon Map: Coding
87(6)
2.11.3 Symbolic Dynamics of the Henon Map: Grammar
93(3)
2.12 Circle Maps
96(4)
2.12.1 A New Global Topology
96(1)
2.12.2 Frequency Locking and Arnold Tongues
96(4)
2.12.3 Chaotic Circle Maps as Limits of Annulus Maps
100(1)
2.13 Annulus Maps
100(4)
2.14 Summary
104(1)
3 Continuous Dynamical Systems: Flows
105(36)
3.1 Definition of Dynamical Systems
105(1)
3.2 Existence and Uniqueness Theorem
106(1)
3.3 Examples of Dynamical Systems
107(13)
3.3.1 Duffing Equation
107(2)
3.3.2 Van der Pol Equation
109(2)
3.3.3 Lorenz Equations
111(2)
3.3.4 Rossler Equations
113(1)
3.3.5 Examples of Nondynamical Systems
114(1)
3.3.5.1 Equation with Non-Lipschitz Forcing Terms
115(1)
3.3.5.2 Delay Differential Equations
115(1)
3.3.5.3 Stochastic Differential Equations
116(1)
3.3.6 Additional Observations
117(3)
3.4 Change of Variables
120(5)
3.4.1 Diffeomorphisms
120(1)
3.4.2 Examples
121(3)
3.4.3 Structure Theory
124(1)
3.5 Fixed Points
125(6)
3.5.1 Dependence on Topology of Phase Space
125(1)
3.5.2 How to Find Fixed Points in Rn
126(1)
3.5.3 Bifurcations of Fixed Points
127(3)
3.5.4 Stability of Fixed Points
130(1)
3.6 Periodic Orbits
131(3)
3.6.1 Locating Periodic Orbits in Rn-1 × S1
131(1)
3.6.2 Bifurcations of Fixed Points
132(1)
3.6.3 Stability of Fixed Points
133(1)
3.7 Flows Near Nonsingular Points
134(2)
3.8 Volume Expansion and Contraction
136(1)
3.9 Stretching and Squeezing
137(1)
3.10 The Fundamental Idea
138(1)
3.11 Summary
139(2)
4 Topological Invariants
141(34)
4.1 Stretching and Squeezing Mechanisms
141(4)
4.2 Linking Numbers
145(14)
4.2.1 Definitions
146(1)
4.2.2 Reidemeister Moves
147(1)
4.2.3 Braids
148(3)
4.2.4 Examples
151(2)
4.2.5 Linking Numbers for a Horseshoe
153(1)
4.2.6 Linking Numbers for the Lorenz Attractor
154(1)
4.2.7 Linking Numbers for the Period-Doubling Cascade
154(1)
4.2.8 Local Torsion
155(1)
4.2.9 Writhe and Twist
156(2)
4.2.10 Additional Properties
158(1)
4.3 Relative Rotation Rates
159(10)
4.3.1 Definition
160(1)
4.3.2 Computing Relative Rotation Rates
160(3)
4.3.3 Horseshoe Mechanism
163(5)
4.3.4 Additional Properties
168(1)
4.4 Relation between Linking Numbers and Relative Rotation Rates
169(1)
4.5 Additional Uses of Topological Invariants
170(4)
4.5.1 Bifurcation Organization
170(1)
4.5.2 Torus Orbits
171(1)
4.5.3 Additional Remarks
171(3)
4.6 Summary
174(1)
5 Branched Manifolds
175(52)
5.1 Closed Loops
175(3)
5.2 What Does This Have to Do with Dynamical Systems?
178(1)
5.3 General Properties of Branched Manifolds
178(3)
5.4 Birman-Williams Theorem
181(3)
5.4.1 Birman-Williams Projection
182(1)
5.4.2 Statement of the Theorem
183(1)
5.5 Relaxation of Restrictions
184(2)
5.5.1 Strongly Contracting Restriction
184(1)
5.5.2 Hyperbolic Restriction
185(1)
5.6 Examples of Branched Manifolds
186(8)
5.6.1 Smale-Rossler System
186(2)
5.6.2 Lorenz System
188(1)
5.6.3 Duffing System
189(3)
5.6.4 Van der Pol System
192(2)
5.7 Uniqueness and Nonuniqueness
194(6)
5.7.1 Local Moves
195(2)
5.7.2 Global Moves
197(3)
5.8 Standard Form
200(1)
5.9 Topological Invariants
201(6)
5.9.1 Kneading Theory
202(3)
5.9.2 Linking Numbers
205(2)
5.9.3 Relative Rotation Rates
207(1)
5.10 Additional Properties
207(9)
5.10.1 Period as Linking Number
208(1)
5.10.2 EBK-Like Expression for Periods
208(1)
5.10.3 Poincare Section
209(1)
5.10.4 Blow-Up of Branched Manifolds
210(1)
5.10.5 Branched-Manifold Singularities
211(1)
5.10.6 Constructing a Branched Manifold from a Map
212(1)
5.10.7 Topological Entropy
213(3)
5.11 Subtemplates
216(8)
5.11.1 Two Alternatives
216(2)
5.11.2 A Choice
218(1)
5.11.3 Topological Entropy
219(2)
5.11.4 Subtemplates of the Smale Horseshoe
221(1)
5.11.5 Subtemplates Involving Tongues
222(2)
5.12 Summary
224(3)
6 Topological Analysis Program
227(44)
6.1 Brief Summary of the Topological Analysis Program
227(1)
6.2 Overview of the Topological Analysis Program
228(6)
6.2.1 Find Periodic Orbits
228(1)
6.2.2 Embed in R3
229(1)
6.2.3 Compute Topological Invariants
230(1)
6.2.4 Identify Template
230(1)
6.2.5 Verify Template
231(1)
6.2.6 Model Dynamics
232(1)
6.2.7 Validate Model
233(1)
6.3 Data
234(9)
6.3.1 Data Requirements
235(1)
6.3.2 Processing in the Time Domain
236(2)
6.3.3 Processing in the Frequency Domain
238(1)
6.3.3.1 High-Frequency Filter
238(1)
6.3.3.2 Low-Frequency Filter
238(1)
6.3.3.3 Derivatives and Integrals
239(1)
6.3.3.4 Hilbert Transforms
240(1)
6.3.3.5 Fourier Interpolation
241(1)
6.3.3.6 Transform and Interpolation
242(1)
6.4 Embeddings
243(13)
6.4.1 Embeddings for Periodically Driven Systems
244(1)
6.4.2 Differential Embeddings
244(3)
6.4.3 Differential-Integral Embeddings
247(1)
6.4.4 Embeddings with Symmetry
248(1)
6.4.5 Time-Delay Embeddings
249(2)
6.4.6 Coupled-Oscillator Embeddings
251(1)
6.4.7 SVD Projections
252(2)
6.4.8 SVD Embeddings
254(1)
6.4.9 Embedding Theorems
254(2)
6.5 Periodic Orbits
256(6)
6.5.1 Close Returns Plots for Flows
256(2)
6.5.1.1 Close Returns Histograms
258(1)
6.5.1.2 Tests for Chaos
258(1)
6.5.2 Close Returns in Maps
259(1)
6.5.2.1 First Return Map
259(1)
6.5.2.2 pth Return Map
260(1)
6.5.3 Metric Methods
261(1)
6.6 Computation of Topological Invariants
262(1)
6.6.1 Embed Orbits
262(1)
6.6.2 Linking Numbers and Relative Rotation Rates
262(1)
6.6.3 Label Orbits
263(1)
6.7 Identify Template
263(1)
6.7.1 Period-1 and Period-2 Orbits
263(1)
6.7.2 Missing Orbits
264(1)
6.7.3 More Complicated Branched Manifolds
264(1)
6.8 Validate Template
264(1)
6.8.1 Predict Additional Toplogical Invariants
265(1)
6.8.2 Compare
265(1)
6.8.3 Global Problem
265(1)
6.9 Model Dynamics
265(3)
6.10 Validate Model
268(2)
6.10.1 Qualitative Validation
269(1)
6.10.2 Quantitative Validation
269(1)
6.11 Summary
270(1)
7 Folding Mechanisms: A2
271(66)
7.1 Belousov-Zhabotinskii Chemical Reaction
272(13)
7.1.1 Location of Periodic Orbits
273(1)
7.1.2 Embedding Attempts
274(4)
7.1.3 Topological Invariants
278(3)
7.1.4 Template
281(1)
7.1.5 Dynamical Properties
282(1)
7.1.6 Models
283(1)
7.1.7 Model Verification
283(2)
7.2 Laser with Saturable Absorber
285(3)
7.3 Stringed Instrument
288(6)
7.3.1 Experimental Arrangement
288(2)
7.3.2 Flow Models
290(1)
7.3.3 Dynamical Tests
291(1)
7.3.4 Topological Analysis
291(3)
7.4 Lasers with Low-Intensity Signals
294(3)
7.4.1 SVD Embedding
295(1)
7.4.2 Template Identification
296(1)
7.4.3 Results of the Analysis
297(1)
7.5 The Lasers in Lille
297(25)
7.5.1 Class B Laser Model
298(6)
7.5.2 CO2 Laser with Modulated Losses
304(4)
7.5.3 Nd-Doped YAG Laser
308(3)
7.5.4 Nd-Doped Fiber Laser
311(7)
7.5.5 Synthesis of Results
318(4)
7.6 The Laser in Zaragoza
322(6)
7.7 Neuron with Subthreshold Oscillations
328(6)
7.8 Summary
334(3)
8 Tearing Mechanisms: A3
337(20)
8.1 Lorenz Equations
337(6)
8.1.1 Fixed Points
338(1)
8.1.2 Stability of Fixed Points
339(1)
8.1.3 Bifurcation Diagram
339(2)
8.1.4 Templates
341(2)
8.1.5 Shimizu-Morioka Equations
343(1)
8.2 Optically Pumped Molecular Laser
343(9)
8.2.1 Models
344(2)
8.2.2 Amplitudes
346(1)
8.2.3 Template
346(1)
8.2.4 Orbits
347(3)
8.2.5 Intensities
350(2)
8.3 Fluid Experiments
352(2)
8.3.1 Data
352(1)
8.3.2 Template
353(1)
8.4 Why A3?
354(1)
8.5 Summary
354(3)
9 Unfoldings
357(34)
9.1 Catastrophe Theory as a Model
357(5)
9.1.1 Overview
357(1)
9.1.2 Example
358(1)
9.1.3 Reduction to a Germ
359(2)
9.1.4 Unfolding the Germ
361(1)
9.1.5 Summary of Concepts
362(1)
9.2 Unfolding of Branched Manifolds: Branched Manifolds as Germs
362(3)
9.2.1 Unfolding of Folds
362(1)
9.2.2 Unfolding of Tears
363(2)
9.3 Unfolding within Branched Manifolds: Unfolding of the Horseshoe
365(10)
9.3.1 Topology of Forcing: Maps
365(1)
9.3.2 Topology of Forcing: Flows
366(3)
9.3.3 Forcing Diagrams
369(2)
9.3.3.1 Orbits with Zero Entropy
371(1)
9.3.3.2 Orbits with Positive Entropy
372(1)
9.3.3.3 Additional Comments
372(2)
9.3.4 Basis Sets of Orbits
374(1)
9.3.5 Coexisting Basins
375(1)
9.4 Missing Orbits
375(2)
9.5 Routes to Chaos
377(1)
9.6 Orbit Forcing and Topological Entropy: Mathematical Aspects
378(5)
9.6.1 General Outline
378(1)
9.6.2 Basic Mathematical Concepts
379(1)
9.6.2.1 Braids and Braid Types
379(1)
9.6.2.2 Braids and Surface Homeomorphisms
380(1)
9.6.2.3 Nielsen-Thurston Classification
381(1)
9.6.2.4 Application to Periodic Orbits and Braid Types
382(1)
9.7 Topological Measures of Chaos in Experiments
383(6)
9.7.1 Mixing in Fluids
383(2)
9.7.2 Chaos in an Optical Parametric Oscillator
385(4)
9.8 Summary
389(2)
10 Symmetry
391(38)
10.1 Information Loss and Gain
391(2)
10.1.1 Information Loss
391(1)
10.1.2 Exchange of Symmetry
392(1)
10.1.3 Information Gain
392(1)
10.1.4 Symmetries of the Standard Systems
392(1)
10.2 Cover and Image Relations
393(1)
10.2.1 General Setup
393(1)
10.3 Rotation Symmetry 1: Images
394(6)
10.3.1 Image Equations and Flows
394(2)
10.3.2 Image of Branched Manifolds
396(2)
10.3.3 Image of Periodic Orbits
398(2)
10.4 Rotation Symmetry 2: Covers
400(4)
10.4.1 Topological Index
401(1)
10.4.2 Covers of Branched Manifolds
402(1)
10.4.3 Covers of Periodic Orbits
403(1)
10.5 Peeling: a New Global Bifurcation
404(3)
10.5.1 Orbit Perestroika
405(1)
10.5.2 Covering Equations
405(2)
10.6 Inversion Symmetry: Driven Oscillators
407(2)
10.7 Duffing Oscillator
409(4)
10.8 Van der Pol Oscillator
413(5)
10.9 Summary
418(11)
11 Bounding Tori
429(8)
11.1 Stretching & Folding vs. Tearing & Squeezing
420(1)
11.2 Inflation
421(1)
11.3 Boundary of Inflation
422(1)
11.4 Index
423(1)
11.5 Projection
424(2)
11.6 Nature of Singularities
426(1)
11.7 Trinions
427(2)
11.8 Poincare Surface of Section
429(1)
11.9 Construction of Canonical Forms
429(3)
11.10 Perestroikas
432(3)
11.10.1 Enlarging Branches
433(1)
11.10.2 Starving Branches
433(2)
11.11 Summary
435(2)
12 Representation Theory for Strange Attractors
437(20)
12.1 Embeddings, Representations, Equivalence
438(1)
12.2 Simplest Class of Strange Attractors
439(1)
12.3 Representation Labels
440(6)
12.3.1 Parity
440(1)
12.3.2 Global Torsion
441(4)
12.3.3 Knot Type
445(1)
12.4 Equivalence of Representations with Increasing Dimension
446(4)
12.4.1 Parity
447(1)
12.4.2 Knot Type
447(1)
12.4.3 Global Torsion
448(2)
12.5 Genus-g Attractors
450(1)
12.6 Representation Labels
451(2)
12.6.1 Parity
451(1)
12.6.2 Multitorsion Index
451(1)
12.6.3 Knot Type
452(1)
12.7 Equivalence in Increasing Dimension
453(2)
12.7.1 Parity and Knot Type
453(1)
12.7.2 Multitorsion Index
453(2)
12.8 Summary
455(2)
13 Flows in Higher Dimensions
457(36)
13.1 Review of Classification Theory in R3
457(2)
13.2 General Setup
459(3)
13.2.1 Spectrum of Lyapunov Exponents
459(2)
13.2.2 Double Projection
461(1)
13.3 Flows in R4
462(4)
13.3.1 Cyclic Phase Spaces
462(1)
13.3.2 Floppiness and Rigidity
462(1)
13.3.3 Singularities in Return Maps
463(3)
13.4 Cusps in Weakly Coupled, Strongly Dissipative Chaotic Systems
466(4)
13.4.1 Coupled Logistic Maps
466(3)
13.4.2 Coupled Diode Resonators
469(1)
13.5 Cusp Bifurcation Diagrams
470(5)
13.5.1 Cusp Return Maps
472(1)
13.5.2 Structure in the Control Plane
472(2)
13.5.3 Comparison with the Fold
474(1)
13.6 Nonlocal Singularities
475(2)
13.6.1 Multiple Cusps
475(2)
13.7 Global Boundary Conditions
477(4)
13.8 From Braids to Triangulations: toward a Kinematics in Higher Dimensions
481(9)
13.8.1 Knot Theory in Three Dimensions and Beyond
481(1)
13.8.2 From Nonintersection to Orientation Preservation
482(8)
13.8.3 Singularities in Higher Dimensions
490(1)
13.9 Summary
490(3)
14 Program for Dynamical Systems Theory
493(20)
14.1 Reduction of Dimension
494(2)
14.2 Equivalence
496(1)
14.3 Structure Theory
497(1)
14.3.1 Reducibility of Dynamical Systems
497(1)
14.4 Germs
498(2)
14.5 Unfolding
500(2)
14.6 Paths
502(1)
14.7 Rank
502(2)
14.7.1 Stretching and Squeezing
503(1)
14.8 Complex Extensions
504(1)
14.9 Coxeter-Dynkin Diagrams
504(2)
14.10 Real Forms
506(1)
14.11 Local vs. Global Classification
507(1)
14.12 Cover-Image Relations
508(1)
14.13 Symmetry Breaking and Restoration
508(3)
14.13.1 Entrainment and Synchronization
509(2)
14.14 Summary
511(2)
Appendix A Determining Templates from Topological Invariants
513(28)
A.1 The Fundamental Problem
513(2)
A.2 From Template Matrices to Topological Invariants
515(1)
A.2.1 Classification of Periodic Orbits by Symbolic Names
515(1)
A.2.2 Algebraic Description of a Template
516(1)
A.2.3 Local Torsion
517(1)
A.2.4 Relative Rotation Rates: Examples
517(2)
A.2.5 Relative Rotation Rates: General Case
519(4)
A.3 Identifying Templates from Invariants
523(1)
A.3.1 Using an Independent Symbolic Coding
524(3)
A.3.2 Simultaneous Determination of Symbolic Names and Template
527(4)
A.4 Constructing Generating Partitions
531(1)
A.4.1 Symbolic Encoding as an Interpolation Process
531(4)
A.4.2 Generating Partitions for Experimental Data
535(1)
A.4.3 Comparison with Methods Based on Homoclinic Tangencies
536(2)
A.4.4 Symbolic Dynamics on Three Symbols
538(1)
A.5 Summary
539(2)
Appendix B Embeddings
541(24)
B.1 Diffeomorphisms
541(2)
B.2 Mappings of Data
543(1)
B.2.1 Too Little Data
543(2)
B.2.2 Too Much Data
545(2)
B.2.3 Just the Right Amount of Data
547(1)
B.3 Tests for Embeddings
547(2)
B.4 Tests of Embedding Tests
549(1)
B.4.1 Trial Data Set
549(1)
B.5 Geometric Tests for Embeddings
550(1)
B.5.1 Fractal Dimension Estimation
550(3)
B.5.2 False Near Neighbor Estimates
553(1)
B.6 Dynamical Tests for Embeddings
554(1)
B.7 Topological Test for Embeddings
555(2)
B.8 Postmortem on Embedding Tests
557(1)
B.8.1 Generality
557(1)
B.8.2 Computational Load
557(1)
B.8.3 Variability
558(1)
B.8.4 Statistics
559(1)
B.8.5 Parameters
560(1)
B.8.6 Noise
560(1)
B.8.7 The Self-Intersection Problem
561(1)
B.8.8 Reliability and Limitations
561(1)
B.9 Stationarity
562(1)
B.10 Beyond Embeddings
563(1)
B.11 Summary
563(2)
Appendix C Frequently Asked Questions
565(4)
C.1 Is Template Analysis Valid for Non-Hyperbolic Systems?
565(1)
C.2 Can Template Analysis Be Applied to Weakly Dissipative Systems?
566(1)
C.3 What About Higher-Dimensional Systems?
567(2)
References 569(12)
Index 581
ROBERT GILMORE, PhD, is a professor in the Physics Department of Drexel University, Philadelphia, Pennsylvania. MARC LEFRANC, PhD, is a researcher at the Centre National de la Recherche Scientifique in the Laboratoire de Physique des Lasers, Atomes, Molecules at the Universite des Sciences et Technologies de Lille, France. The authors are internationally recognized leaders in the field who have been developing these techniques for about two decades. As active members in the community they are knowledgeable about the broader context of their book's subject.
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