Homoclinic orbit

In the study of dynamical systems, a homoclinic orbit is a path through phase space which joins a saddle equilibrium point to itself. More precisely, a homoclinic orbit lies in the intersection of the stable manifold and the unstable manifold of an equilibrium. It is a heteroclinic orbit–a path between any two equilibrium points–in which the endpoints are one and the same.

Consider the continuous dynamical system described by the ordinary differential equation

${\displaystyle {\dot {x}}=f(x)}$

Suppose there is an equilibrium at ${\displaystyle x=x_{0}}$, then a solution ${\displaystyle \Phi (t)}$ is a homoclinic orbit if

${\displaystyle \Phi (t)\rightarrow x_{0}\quad \mathrm {as} \quad t\rightarrow \pm \infty }$

If the phase space has three or more dimensions, then it is important to consider the topology of the unstable manifold of the saddle point. The figures show two cases. First, when the stable manifold is topologically a cylinder, and secondly, when the unstable manifold is topologically a Möbius strip; in this case the homoclinic orbit is called twisted.