Exotic Attractors: From Liapunov Stability to Riddled Basins

ISBN-13: 9783034874236 / Angielski / Miękka / 2012 / 130 str.

Jorge Buescu

Opis

This book grew out of the work developed at the University of Warwick, under the supervision of Ian Stewart, which formed the core of my Ph.D. Thesis. Most of the results described were obtained in joint work with Ian; as usual under these circumstances, many have been published in research journals over the last two years. Part of Chapter 3 was also joint work with Peter Ashwin. I would like to stress that these were true collaborations. We worked together at all stages; it is meaningless to try to identify which idea originated from whom. While preparing this book, however, I felt that a mere description of the results would not be fitting. First of all, a book is aimed at a wider audience than papers in research journals. More importantly, the work should assume as little as possible, and it should be brought to a form which is pleasurable, not painful, to read.

Recenzje

"The author gives a thorough insight into topological and ergodic properties of invariant subsets and their structure, classifies the main concepts which are used in the book and describes their characteristics. In addition, some new important concepts which, for the most part, have been previously known only from articles are presented...
The manner of exposition is in the tradition of mathematics: rigorous description of the concepts and notions of attractors in dynamics, and detailed proofs of main results. At the same time, the material is presented at an acceptable level for the wide circle of researchers and post-graduate students who apply ideas of dynamical systems. Moreover, because the book is self-contained, readers can use it as a fine introduction to the modern theory of attractors and related topics.
The book is clearly written, and it has many references (about 120 entries); in addition, some results are supported by helpful examples (and even sometimes counterexamples). Moreover, at the end of each chapter the readers can find very useful general comments and historical remarks."
-- Mathematical Reviews

Spis treści

1 Attractors in Dynamical Systems.- 1.1 Introduction.- 1.2 Basic definitions.- 1.3 Topological and dynamical consequences.- 1.4 Attractors.- 1.5 Examples and counterexamples.- 1.6 Historical remarks and further comments.- 2 Liapunov Stability and Adding Machines.- 2.1 Introduction.- 2.2 Adding Machines and Denjoy maps.- 2.3 Stable Cantor sets are Adding Machines.- 2.4 Adding Machines and periodic points: interval maps.- 2.5 Interlude: Adding Machines as inverse limits.- 2.6 Stable ?-limit sets are Adding Machines.- 2.7 Classification of Adding Machines.- 2.8 Existence of Stable Adding Machines.- 2.9 Historical remarks and further comments.- 3 From Attractor to Chaotic Saddle: a journey through transverse instability.- 3.1 Introduction.- 3.1.1 Riddled and locally riddled basins.- 3.1.2 Symmetry and invariant submanifolds.- 3.2 Normal Liapunov exponents and stability indices.- 3.2.1 Normal Liapunov exponents.- 3.2.2 The normal Liapunov spectrum.- 3.2.3 Stability indices.- 3.2.4 Chaotic saddles.- 3.2.5 Strong SBR measures.- 3.2.6 Classification by normal spectrum.- 3.3 Normal parameters and normal stability.- 3.3.1 Parameter dependence of the normal spectrum.- 3.3.2 Global behaviour near bifurcations.- 3.4 Example: ?2-symmetric maps on ?2.- 3.4.1 The spectrum of normal Liapunov exponents.- 3.4.2 Global transverse stability for f.- 3.4.3 Global transverse stability for g.- 3.5 Example: synchronization of coupled systems.- 3.5.1 Electronic experiments.- 3.5.2 Observations.- 3.5.3 Analysis of the dynamics.- 3.6 Historical remarks and further comments.