Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria

Abstract

We discuss dynamics near homoclinic orbits to saddle-focus equilibria in three-dimensional vector fields. The existence of periodic and strange attractors is investigated not in unfoldings, but in families for which each member has a homoclinic orbit. We consider how often, in the sense of measure, periodic and strange attractors occur in such families. We also discuss the fate of typical orbits, and establish that despite the possible existence of attractors, a large proportion of points from a small vicinity of the homoclinic orbit, lies outside the basin of an attractor.