Since the moment that the concept of chaotic attractor
was proposed, the problem of its characterization arose. Sev-
eral parameters were suggested, some of them are widely
used today such as Lyapunov exponents, fractal dimension,
or entropies, which are useful to quantify the degree of com-
plexity of the chaotic attractor. Nevertheless, the information
provided by these parameters is reduced because they do not
describe the attractor structure. In order to do so, an object
suitable to characterize the attractor ?the template? and a pro-
cedure to find this object have been proposed for systems
whose dynamics can be modeled by a three-dimensional
phase space or which rapidly relax to a three-dimensional
subspace of the phase space. Both the object and the proce-
dure are based on the Birman-Williams theorem ?1,2? which
shows that there is a one-to-one correspondence between the
periodic orbits in flows in R2?S1having a contracting direc-
tion and the orbits in a branched manifold which can be
thought of as the “limit for infinite contracting rate” of the
flow. Such a branched manifold is called “template” or “knot
holder.” In general, a template may be a very complex object
?see ?3? for a thorough template classification?, but templates
required to describe physical phenomena are much simpler,
as explained in Sec. II. They can be fully characterized by
means of integer numbers, which allow one to get a clear
answer with regard to the validity of a theoretical model
candidate to account for an experimental time series: if the
analysis of the theoretical and the experimental time series
provides different templates, the model is not valid, at least
with the parameters chosen for the simulation ?when dealing
with real parameters, frequently the answer is not that clear:
whether the model is valid or not depends on the tolerable
difference between the theoretical and experimental values
of the parameters chosen; in the end, tolerance is a subjective
matter?.

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