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 ODE
Suppose there are equilibria at formula_2 and formula_3, then a solution formula_4 is a heteroclinic orbit from formula_5 to formula_6 if
and
This implies that the orbit is contained in the stable manifold of formula_6 and the unstable manifold of formula_5.
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 formula_11 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 formula_14 is a sequence of symbols of length "k", (of course, formula_15), and formula_16 is another sequence of symbols, of length "m" (likewise, formula_17). The notation formula_18 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 formula_20 being non-empty, and, of course, not being "p", as otherwise, the orbit would simply be formula_18.