Homoclinic Explosions: The First Homoclinic Explosion

Abstract

When r > 1 there is a two-dimensional sheet of initial values in R³ from which trajectories tend towards the origin. This two-dimensional sheet is called the stable manifold of the origin. Near the origin we know that this sheet looks like a plane (the plane associated with the two negative eigenvalues of the flow linearized near the origin) and if we wished we could approximate its position elsewhere by integrating the equations in backwards time with initial conditions lying on this plane and near to the origin. It appears that when r is only moderately larger than one, the stable manifold of the origin divides R³ into two halves in a fairly simple way. Trajectories started in one half-space tend towards C₁ and trajectories started in the other half-space tend towards C₂. Trajectories started on the stable manifold of the origin tend, of course, towards the origin. (See Fig. 2.1.)