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::$dot\; x=f(x)$Suppose there are equilibria at $x=x\_0$ and $x=x\_1$, then a solution $phi(t)$ is a heteroclinic orbit from $x\_0$ to $x\_1$ if::$phi(t)\; ightarrow\; x\_0quad\; mathrm\{as\}quad\; t\; ightarrow-infty$and::$phi(t)\; ightarrow\; x\_1quad\; mathrm\{as\}quad\; t\; ightarrow+infty$

This implies that the orbit is contained in the stable manifold of $x\_1$ and the unstable manifold of $x\_0$.

See also

* Heteroclinic cycle
* Heteroclinic bifurcation
* Homoclinic orbit

References

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