The Topology of Chaos is a highly valued resource for those who wish to move from an introductory knowledge of chaotic behavior to a more sophisticated and precise understanding of chaotic systems and measurements made on them. The authors provide deep insight into the structure of strange attractors, how they are classified, and how the information required to characterize a strange attractor can be extracted from experimental data. What makes this book special is the abundance of practical examples where time series from real physical systems (e.g. lasers) are analyzed using topological techniques. Hence it has become the experimenter’s guidebook to reliable studies of experimental data for comparison with candidate theoretical models, invaluable to physicists, mathematicians, and engineers studying low-dimensional chaotic systems.
This second edition incorporates recent advances in the field, such as the concept of bounding tori.
From the contents:
- Discrete Dynamical Systems: Maps
- Continuous Dynamical Systems: Flows
- Topological Invariants
- Branched Manifolds
- Topological Analysis Program
- Folding Mechanisms: A2
- Tearing Mechanisms: A3
- Bounding Tori
- Representation Theory for Strange Attractors
- Flows in Higher Dimensions
- Program for Dynamical System Theory
Table of contents
2. Discrete 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
11. Bounding Tori
12. Representation Theory for Strange Attractors
13. Flows in Higher Dimensions
14. Program for Dynamical System Theory
Appendix A: Determining Templates from Topological Invariants
Appendix B: Embeddings
New To This Edition
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.
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)