[[Image:Heteroclinic orbit in pendulum phaseportrait.png|thumb|right|The [[phase portrait]] of the pendulum equation x″ + sin x = 0 . The highlighted curve shows the heteroclinic orbit from (x, x′) = (–π, 0) to (x, x′) = (π, 0) . This orbit corresponds with the (rigid) pendulum starting upright, making one revolution through its lowest position, and ending upright again.]] 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 x=x_0,x_1. Then a solution \phi(t) is a heteroclinic orbit from x_0 to x_1 if both limits are satisfied: This implies that the orbit is contained in the stable manifold of x_1 and the unstable manifold of x_0.

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, (of course, t_i\in S), and is another sequence of symbols, of length m (likewise, r_i\in S). The notation p^\omega 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 p^\omega.