Homoclinic bifurcation

Summary

A homoclinic bifurcation is a global bifurcation in which, at one parameter value, an orbit leaves a saddle equilibrium and returns to it. This orbit is a homoclinic orbit. As the parameter crosses this value, the loop breaks, and periodic orbits can be created or destroyed near it. In the plane, under nondegeneracy conditions, a single limit cycle is born from the loop. In three dimensions, near a saddle-focus, the bifurcation can produce chaos, as the Shilnikov theorem shows.