In mathematics, a heteroclinic cycle is an invariant set in the phase space of a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

. It is a topological circle of equilibrium points and connecting heteroclinic orbit

Heteroclinic orbit

In mathematics, in the phase portrait of a dynamical system, a heteroclinic orbit is a path in phase space which joins two different equilibrium points...

s. 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. An prototypical example of a robust heteroclinic cycle is the Guckenheimer–Holmes cycle.