Symbolic Dynamics
By using the Markov partition, the long-time behaviour of hyperbolic system can be studied using the techniques of symbolic dynamics. In this case, a homoclinic orbit has a particularly simple and clear representation. Suppose that is a finite set of M symbols. The dynamics of a point x is then represented by a bi-infinite string of symbols
A periodic point of the system is simply a recurring sequence of letters. A heteroclinic orbit is then the joining of two distinct periodic orbits. It may be written as
where is a sequence of symbols of length k, (of course, ), and is another sequence of symbols, of length m (likewise, ). The notation simply denotes the repetition of p an infinite number of times. Thus, a heteroclinic orbit can be understood as the transition from one periodic orbit to another. By contrast, a homoclinic orbit can be written as
with the intermediate sequence being non-empty, and, of course, not being p, as otherwise, the orbit would simply be .
Read more about this topic: Homoclinic Orbit
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