# 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 and , then a solution is a heteroclinic orbit from to if

and

This implies that the orbit is contained in the stable manifold of and the unstable manifold of .

## 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, ), and is another sequence of symbols, of length *m* (likewise, ). The notation 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 .

## See also

## References

- John Guckenheimer and Philip Holmes,
*Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*, (Applied Mathematical Sciences Vol.**42**), Springer