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The Topology of Chaos: Alice in Stretch and Squeezeland
R. Gilmore and M. Lefranc
Wiley Interscience Seires, (2002) 520 pp.,
price USD 120.-
ISBN:0471408166
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Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.
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Level: beginning/intermediate.
This book presents itself as an introduction to the topological
analysis approach to dynamical systems, as envisioned by the
authors and their collaborators in the physics community.
This approach to dynamical systems is briefly summarized as follows.
There are a number of dynamical systems that, correctly or not, can
be interpreted as a strange attractor for a flow in
three-dimensional Euclidean space. For such a flow, the strange
attractor can be `approximated' as an embedded branched surface. The
canonical example of this is the Lorenz attractor (Lorenz'
original paper remarks on this resemblance). The entire strange
attractor, with all its attendant periodic orbits, can be described
by a few pieces of data which encode the knotting and twisting of
the surface and the combinatorics of the branching. If one wants to
characterize a complicated three-dimensional flow by a simple set of
invariants, then understanding the topology of these branched
manifolds (a.k.a. `templates') is a useful endeavor. In
particular, many of the systems considered in this text come from a
Takens embedding of experimental time-series data, in which case
there are a number of interesting issues associated with identifying
the `minimal model' of strange attractor present in the data.
As everyone knows, the Lorenz attractor that appears on the computer
screen is not really a branched surface; however, as pointed out by
Guckenheimer and Williams in the late 1970s, it is possible to
approximate it as such and use this approximation to great effect.
Neither is the Lorenz attractor uniformly hyperbolic -- a fact that
has caused no small amount of grief to mathematicians, while sparing
much of the rest of the world. In like manner, Gilmore and Lefranc
do not waste time, space, or effort worrying about the details of
the dynamics. The goal is to approximate everything in sight with a
strange attractor, and then characterize its embedding data
succinctly.
This text exhibits an enthusiasm for its subject that, at its
zenith, promotes the topological analysis of branched manifolds as a
key step in the 13-part model of what the authors describe as a
proposed "mature" reformulation of dynamical systems theory
(cross-linked with similar components of Lie theory and singularity
theory). There does not seem to be much recognition of the
sad-but-true fact that three-dimensional flows --- much like their
cousins the 1-d maps and 2-d homeomorphisms --- ultimately comprise
a small albeit fascinating corner of dynamical systems.
This book makes up for its lack of precision and rigor with an
abundance of interesting physically relevant examples. The figures
are numerous and illustrative. The writing style of the book is
foreshadowed by its title (though, unfortunately, there are no
additional references in the text to Alice or to any other
interesting denizens). The book is probably not appropriate as a
textbook, being rather a breezy introduction to the subject.