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

{\dot {x}}=f(x).

Suppose there are equilibria at $x=x_{0},x_{1}.$ Then a solution $\phi (t)$ is a heteroclinic orbit from $x_{0}$ to $x_{1}$ if both limits are satisfied:

{\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 $x_{1}$ and the unstable manifold of $x_{0}$ .

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