The Topology of Chaos: Alice in Stretch and Squeezeland

by Robert Gilmore, Marc Lefranc

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
* Discrete Dynamical Maps
* Continuous Dynamical 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.

518 pages, Hardcover

First published January 1, 2002

Reviews

[C. Hinsley · Rating 5 out of 5]

This book is the primary resource on advanced methods in experimental and theoretical chaos topology, introducing Birman-Williams projection theory, templates, and a novel approach to topological data analysis. Gives ample preparation for the template renormalization theory of Ghrist, Holmes, & Sullivan as well as the recent homological templex theory of Sciamarella. Lots of applications in the book.

[Tommy Powell · Rating 3 out of 5]

I simply don't know enough math to do any more than graze upon this work. However, I find the explicit focus on "small parts" of chaotic signatures very, very compelling.