The topology of chaos. Alice in stretch and Squeezeland. (English) Zbl 1019.37016

Chichester: Wiley. xxiv, 495 p. (2002).
The subject of this very nice book is the analysis of data generated by dynamical systems that exhibit chaotic behavior, and the main goal is to develop a new topological analysis procedure which helps to extract from chaotic data information useful for understanding chaotic dynamical systems and their evolution. To this end, invariants of periodic orbits and their Gaussian linking numbers are studied, and a refined topological invariant based on periodic orbits, the relative rotation rates, is introduced. These invariants are used to identify topological structures, branched manifolds, which are used for classifying strange attractors “in the large.” As result, “low-dimensional” strange attractors that could be embedded in three-dimensional spaces are classified. The classification is topological and depends on the organization of periodic orbits “in” the strange attractor. Furthermore, the topological analysis algorithm comes with the first and the only chaotic data analysis procedure with reject/fail to reject test criteria. Finally, a secondary, more refined classification for strange attractors, depending on identifying a branched manifold which describes the stretching and squeezing mechanisms that generate strange attractors, and a basis set of orbits which describes the spectrum of all the unstable periodic orbits “in” a strange attractor, is introduced.
The book consists of twelve chapters and an appendix. The first chapter introduces the reader to the subject, outlines the basic objectives and summarizes the accomplishments of the new topological analysis method. Chapter two contains many results on discrete dynamical systems useful for the topological analyses that follow. Important properties of continuous dynamical systems are reviewed in Chapter three. Topological invariants are discussed in Chapter four, where Gaussian linking numbers between pairs of orbits are introduced for the study of chaotic dynamical systems. Branched manifolds, geometric structures that support all the unstable periodic orbits in the strange attractor, are introduced in Chapter five. A step-by-step account of the topological analysis method is presented in Chapter six. Different squeezing and stretching mechanisms are discussed in Chapters seven to nine – the simplest catastrophes, the fold \(A_2\) and the cusp \(A_3,\) and the unfoldings of the “germs” of dynamical systems. Elegant and powerful results relating cover and image dynamical systems, even when the symmetry group relating them has only two group elements, form the subject of Chapter ten. Several interesting results on four-dimensional chaotic flows with two unstable directions and one stable direction are discussed in Chapter eleven prompting future developments in the field. Chapter twelve compares Lie group theory, singularity theory, and dynamical systems theory suggesting how some answers might expedite the development of the latter one.
Finally, computational techniques useful for determining the simplest template compatible with a given set of topological invariants are described in the Appendix, where these techniques have been also implemented in computer programs. There are more than two hundred references, most published within the last fifteen years, which are helpful for further explorations of the subject.
The book is well written, with rigorous and clear exposition of the material, and is pleasant to read. It casts light on many important issues related to the topological properties of “strange attractors” and will be very useful both for researchers and practitioners working in dynamical systems, as well as for graduate students in mathematics and physics.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37Cxx Smooth dynamical systems: general theory
65P20 Numerical chaos
54H20 Topological dynamics (MSC2010)
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
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