|
" The topology of chaos : "
Robert Gilmore, Marc Lefranc.
| Document Type
|
:
|
BL
|
| Record Number
|
:
|
1025871
|
| Doc. No
|
:
|
b780241
|
| Main Entry
|
:
|
Gilmore, Robert,1941-
|
| Title & Author
|
:
|
The topology of chaos : : Alice in stretch and squeezeland /\ Robert Gilmore, Marc Lefranc.
|
| Publication Statement
|
:
|
New York :: Wiley-Interscience,, ©2002.
|
| Page. NO
|
:
|
xxiii, 495 pages :: illustrations ;; 25 cm
|
| ISBN
|
:
|
0471408166
|
|
|
:
|
: 9780471408161
|
| Bibliographies/Indexes
|
:
|
Includes bibliographical references and index.
|
| Contents
|
:
|
1. Introduction. 1.1. Laser with Modulated Losses. 1.2. Objectives of a New Analysis Procedure. 1.3. Preview of Results. 1.4. Organization of This Work -- 2. Discrete Dynamical Systems: Maps. 2.1. Introduction. 2.2. Logistic Map. 2.3. Bifurcation Diagrams. 2.4. Elementary Bifurcations in the Logistic Map. 2.5. Map Conjugacy. 2.6. Fully Developed Chaos in the Logistic Map. 2.7. One-Dimensional Symbolic Dynamics. 2.8. Shift Dynamical Systems, Markov Partitions, and Entropy. 2.9. Fingerprints of Periodic Orbits and Orbit Forcing. 2.10. Two-Dimensional Dynamics: Smale's Horseshoe. 2.11. Henon Map. 2.12. Circle Maps. 2.13. Summary -- 3. Continuous Dynamical Systems: Flows. 3.1. Definition of Dynamical Systems. 3.2. Existence and Uniqueness Theorem. 3.3. Examples of Dynamical Systems. 3.4. Change of Variables. 3.5. Fixed Points. 3.6. Periodic Orbits. 3.7. Flows near Nonsingular Points. 3.8. Volume Expansion and Contraction. 3.9. Stretching and Squeezing. 3.10. The Fundamental Idea. 3.11. Summary -- 4. Topological Invariants. 4.1. Stretching and Squeezing Mechanisms. 4.2. Linking Numbers. 4.3. Relative Rotation Rates. 4.4. Relation between Linking Numbers and Relative Rotation Rates. 4.5. Additional Uses of Topological Invariants. 4.6. Summary -- 5. Branched Manifolds. 5.1. Closed Loops. 5.2. What Has This Got to Do with Dynamical Systems? 5.3. General Properties of Branched Manifolds. 5.4. Birman-Williams Theorem. 5.5. Relaxation of Restrictions. 5.6. Examples of Branched Manifolds. 5.7. Uniqueness and Nonuniqueness. 5.8. Standard Form. 5.9. Topological Invariants. 5.10. Additional Properties. 5.11. Subtemplates. 5.12. Summary -- 6. Topological Analysis Program. 6.1. Brief Summary of the Topological Analysis Program. 6.2. Overview of the Topological Analysis Program. 6.3. Data. 6.4. Embeddings. 6.5. Periodic Orbits. 6.6. Computation of Topological Invariants. 6.7. Identify Template. 6.8. Validate Template. 6.9. Model Dynamics. 6.10. Validate Model. 6.11. Summary -- 7. Folding Mechanisms: A[subscript 2]. 7.1. Belousov-Zhabotinskii Chemical Reaction. 7.2. Laser with Saturable Absorber. 7.3. Stringed Instrument. 7.4. Lasers with Low-Intensity Signals. 7.5. The Lasers in Lille. 7.6. Neuron with Subthreshold Oscillations. 7.7. Summary -- 8. Tearing Mechanisms: A[subscript 3]. 8.1. Lorenz Equations. 8.2. Optically Pumped Molecular Laser. 8.3. Fluid Experiments. 8.4. Why A[subscript 3]? 8.5. Summary -- 9. Unfoldings. 9.1. Catastrophe Theory as a Model. 9.2. Unfolding of Branched Manifolds: Branched Manifolds as Germs. 9.3. Unfolding within Branched Manifolds: Unfolding of the Horseshoe. 9.4. Missing Orbits. 9.5. Routes to Chaos. 9.6. Summary -- 10. Symmetry. 10.1. Information Loss and Gain. 10.2. Cover and Image Relations. 10.3. Rotation Symmetry 1: Images. 10.4. Rotation Symmetry 2: Covers. 10.5. Peeling: A New Global Bifurcation. 10.6. Inversion Symmetry: Driven Oscillators. 10.7. Duffing Oscillator. 10.8. van der Pol Oscillator. 10.9. Summary -- 11. Flows in Higher Dimensions. 11.1. Review of Classification Theory in R[superscript 3]. 11.2. General Setup. 11.3. Flows in R[superscript 4]. 11.4. Cusp Bifurcation Diagrams. 11.5. Nonlocal Singularities. 11.6. Global Boundary Conditions. 11.7. Summary -- 12. Program for Dynamical Systems Theory. 12.1. Reduction of Dimension. 12.2. Equivalence. 12.3. Structure Theory. 12.4. Germs. 12.5. Unfolding. 12.6. Paths. 12.7. Rank. 12.8. Complex Extensions. 12.9. Coxeter-Dynkin Diagrams. 12.10. Real Forms. 12.11. Local vs. Global Classification. 12.12. Cover-Image Relations. 12.13. Symmetry Breaking and Restoration. 12.14. Summary -- App. A. Determining Templates from Topological Invariants.
|
| Abstract
|
:
|
"The behavior of a physical system may appear irregular or chaotic even when it is completely deterministic and predictable for short periods of time into the future. How does one model the dynamics of a system operating in a chaotic regime? Older tools such as estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions do not sufficiently answer this question. In a significant evolution of the field of Nonlinear Dynamics, The Topology of Chaos responds to the fundamental challenge of chaotic systems by introducing a new analysis method - Topological Analysis - which can be used to extract, from chaotic data, the topological signatures that determine the stretching and squeezing mechanisms which act on flows in phase space and are responsible for generating chaotic data." "Suitable at the present time for analyzing "strange attractors" that can be embedded in three-dimensional spaces, this approach offers researchers and practitioners in the discipline a complete and satisfying resolution to the fundamental questions of chaotic systems."--Jacket.
|
| Subject
|
:
|
Attractors (Mathematics)
|
| Subject
|
:
|
Chaotic behavior in systems.
|
| Subject
|
:
|
Attracteurs (Mathématiques)
|
| Subject
|
:
|
Chaos.
|
| Subject
|
:
|
Attractors (Mathematics)
|
| Subject
|
:
|
Chaostheorie
|
| Subject
|
:
|
Chaotic behavior in systems.
|
| Subject
|
:
|
Topologie
|
| Dewey Classification
|
:
|
514/.74
|
| LC Classification
|
:
|
QA614.813.G55 2002
|
| NLM classification
|
:
|
SK 350rvk
|
|
|
:
|
UG 3900rvk
|
| Added Entry
|
:
|
Lefranc, Marc.
|
| |