Consider the continuous dynamical system described by the ordinary differential equation
<math display=block>\dot x = f(x).</math>
Suppose there are equilibria at <math>x=x_0,x_1.</math> Then a solution <math>\phi(t)</math> is a heteroclinic orbit from <math>x_0</math> to <math>x_1</math> if both limits are satisfied:
<math display=block>\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}</math>
This implies that the orbit is contained in the stable manifold of <math>x_1</math> and the unstable manifold of <math>x_0</math>.
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 <math>S=\{1,2,\ldots,M\}</math> is a finite set of M symbols. The dynamics of a point x is then represented by a bi-infinite string of symbols
<math>\sigma =\{(\ldots,s_{-1},s_0,s_1,\ldots) : s_k \in S \; \forall k \in \mathbb{Z} \}</math>
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
<math>p^\omega s_1 s_2 \cdots s_n q^\omega</math>
where <math>p= t_1 t_2 \cdots t_k</math> is a sequence of symbols of length k, (of course, <math>t_i\in S</math>), and <math>q = r_1 r_2 \cdots r_m</math> is another sequence of symbols, of length m (likewise, <math>r_i\in S</math>). The notation <math>p^\omega</math> 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
<math>p^\omega s_1 s_2 \cdots s_n p^\omega</math>
with the intermediate sequence <math>s_1 s_2 \cdots s_n</math> being non-empty, and, of course, not being p, as otherwise, the orbit would simply be <math>p^\omega</math>.
John Guckenheimer and Philip Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, (Applied Mathematical Sciences Vol. 42), Springer
