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 $$\dot x = f(x).$$ 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: $$\begin{array}{rcl} \phi(t) \rightarrow x_0 &\text{as}& t \rightarrow -\infty, \\[4pt] \phi(t) \rightarrow x_1 &\text{as}& t \rightarrow +\infty. \end{array}$$

This implies that the orbit is contained in the stable manifold of $$x_1$$ 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 $$S=\{1,2,\ldots,M\}$$ is a finite set of M symbols. The dynamics of a point x is then represented by a bi-infinite string of symbols


 * $$\sigma =\{(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in S \; \forall k \in \mathbb{Z} \}$$

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


 * $$p^\omega s_1 s_2 \cdots s_n q^\omega$$

where $$p= t_1 t_2 \cdots t_k$$ is a sequence of symbols of length k, (of course, $$t_i\in S$$), and $$q = r_1 r_2 \cdots r_m$$ 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


 * $$p^\omega s_1 s_2 \cdots s_n p^\omega$$

with the intermediate sequence $$s_1 s_2 \cdots s_n$$ being non-empty, and, of course, not being p, as otherwise, the orbit would simply be $$p^\omega$$.