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The Topology of Chaos: Alice in Stretch and Squeezeland

The Topology of Chaos: Alice in Stretch and Squeezeland

Robert Gilmore, Marc Lefranc

ISBN: 978-3-527-61732-6

Sep 2008

518 pages



A new approach to understanding nonlinear dynamics and strange attractors
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. Beginning with an example of a laser that has been operated under conditions in which it behaved chaotically, the authors convey the methodology of Topological Analysis through detailed chapters on:
* Discrete Dynamical Systems: Maps
* Continuous Dynamical Systems: Flows
* Topological Invariants
* Branched Manifolds
* The Topological Analysis Program
* Fold Mechanisms
* Tearing Mechanisms
* Unfoldings
* Symmetry
* Flows in Higher Dimensions
* A Program for Dynamical Systems Theory
Suitable at the present time for analyzing "strange attractors" that can be embedded in three-dimensional spaces, this groundbreaking approach offers researchers and practitioners in the discipline a complete and satisfying resolution to the fundamental questions of chaotic systems.

1. Introduction.

2. Dscrete Dynamical Systems: Maps.

3. Continuous Dynamical Systems: Flows.

4. Topological Invariants.

5. Branched Manifolds.

6. Topological Analysis Program.

7. Folding Mechanisms: A2.

8. Tearing Mechanisms: A3.

9. Unfoldings.

10. Symmetry.

11. Flows in Higher Dimensions.

12. Program for Dynamical Systems Theory.

Appendix A: Determining Templates from Topological Invariants.


Topic Index.
"…an abundance of interesting physically relevant examples. The figures are numerous and illustrative." (Dynamical Systems Magazine, January 2006)

"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)

"...contains a wealth of material and, in particular, many practical examples of how topological information can be extracted from experimental time series." (Mathematical Reviews, 2003k)

"...well written, with rigorous and clear exposition of the material, and is pleasant to read..." (Zentralblatt Math, 2003)