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111 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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