The following model is an example of a three-dimensional vector field exhibiting all types of homoclinic bifurcations. It was developed by Björn Sandstede and in a simplified form, the equations are
We choose fixed parameters a = 0.125, b = 0.875, c = -2, = 1, = 1, and = 3. The parameter is the continuation parameter.
The origin is always an equilibrium. For < 0 a twisted saddle periodic orbit exists that disappears in a twisted homoclinic bifurcation at = 0.The animation below shows the stable manifolds of both this periodic orbit and the origin.
|The animated gif shows the unstable manifold rotating about the z-axis, centered at B (3.4MB).|