Heteroclinic orbit

In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit (sometimes called a heteroclinic connection) 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 ordinary differential equation

Suppose there are equilibria at

is a heteroclinic orbit from

if both limits are satisfied:

{\displaystyle {\begin{array}{rcl}\phi (t)\rightarrow x_{0}&{\text{as}}&t\rightarrow -\infty ,\\[4pt]\phi (t)\rightarrow x_{1}&{\text{as}}&t\rightarrow +\infty .\end{array}}}

This implies that the orbit is contained in the stable manifold of

and the unstable manifold of

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.

is a sequence of symbols of length k, (of course,

is another sequence of symbols, of length m (likewise,

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