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 ordinary differential equation $${\displaystyle {\dot {x}}=f(x).}$$ Suppose there are equilibria at ${\displaystyle x=x_{0},x_{1}.}$ Then a solution ${\displaystyle \phi (t)}$ is a heteroclinic orbit from ${\displaystyle x_{0}}$ to ${\displaystyle x_{1}}$ if both limits are satisfied: $${\displaystyle {\begin{array}{rcl}\phi (t)\rightarrow x_{0}&{\text{as}}&t\rightarrow -\infty ,\\[4pt]\phi (t)\rightarrow x_{1}&{\text{as}}&t\rightarrow +\infty .\end{array}}}$$

This implies that the orbit is contained in the stable manifold of ${\displaystyle x_{1}}$ and the unstable manifold of ${\displaystyle x_{0}}$.