Topological analysis of chaotic dynamical systems

  • Gilmore R
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Topological methods have recently been developed for the analysis
of dissipative dynamical systems that operate in the chaotic regime.
They were originally developed for three-dimensional dissipative
dynamical systems, but they are applicable to all ��low-dimensional��
dynamical systems. These are systems for which the flow rapidly relaxes
to a three-dimensional subspace of phase space. Equivalently, the
associated attractor has Lyapunov dimension dL,3. Topological methods
supplement methods previously developed to determine the values of
metric and dynamical invariants. However, topological methods possess
three additional features: they describe how to model the dynamics;
they allow validation of the models so developed; and the topological
invariants are robust under changes in control-parameter values.
The topological-analysis procedure depends on identifying the stretching
and squeezing mechanisms that act to create a strange attractor and
organize all the unstable periodic orbits in this attractor in a
unique way. The stretching and squeezing mechanisms are represented
by a caricature, a branched manifold, which is also called a template
or a knot holder. This turns out to be a version of the dynamical
system in the limit of infinite dissipation. This topological structure
is identified by a set of integer invariants. One of the truly remarkable
results of the topological-analysis procedure is that these integer
invariants can be extracted from a chaotic time series. Furthermore,
self-consistency checks can be used to confirm the integer values.
These integers can be used to determine whether or not two dynamical
systems are equivalent; in particular, they can determine whether
a model developed from time-series data is an accurate representation
of a physical system. Conversely, these integers can be used to provide
a model for the dynamical mechanisms that generate chaotic data.
In fact, the author has constructed a doubly discrete classification
of strange attractors. The underlying branched manifold provides
one discrete classification. Each branched manifold has an ��unfolding��
or perturbation in which some subset of orbits is removed. The remaining
orbits are determined by a basis set of orbits that forces the presence
of all remaining orbits. Branched manifolds and basis sets of orbits
provide this doubly discrete classification of strange attractors.
In this review the author describes the steps that have been developed
to implement the topological-analysis procedure. In addition, the
author illustrates how to apply this procedure by carrying out the
analysis of several experimental data sets. The results obtained
for several other experimental time series that exhibit chaotic behavior
are also described.

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  • R Gilmore

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