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 .