Heteroclinic cycle

In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system. It is a topological circle of equilibrium points and connecting heteroclinic orbits. If a heteroclinic cycle is asymptotically stable, approaching trajectories spend longer and longer periods of time in a neighbourhood of successive equilibria.

Robust heteroclinic cycles

A robust heteroclinic cycle is one which persists under small changes in the underlying dynamical system. Robust cycles often arise in the presence of symmetry or other constraints which force the existence of invariant hyperplanes. A prototypical example of a robust heteroclinic cycle is the Guckenheimer–Holmes cycle.

See also

• Heteroclinic bifurcation
• Heteroclinic network

References

• Guckenheimer J and Holmes, P, 1988, Structurally Stable Heteroclinic Cycles, Math. Proc. Cam. Phil. Soc. 103: 189-192.