Heteroclinic orbit


In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit is a path in phase space which joins two different equilibrium points. If the equilibrium points at the start and end of the orbit are the same, the orbit is a homoclinic orbit.
Consider the continuous dynamical system described by the ODE
Suppose there are equilibria at and, then a solution is a heteroclinic orbit from to if
and
This implies that the orbit is contained in the stable manifold of and the unstable manifold of.

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 heteroclinic 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,, and is another sequence of symbols, of length m. 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.