Homoclinic explosions and implosions

In this paper we study the interaction of local saddle node and Hopf bifurcations with global recurrent orbits. We consider a four-parameter family of three-dimensional vector fields which are small perturbations of an integrable system possessing a line of degenerate saddle points connected by a manifold of homoclinic loops. Our most striking finding is that homoclinic bifurcations occur in which a unique connecting orbit is replaced by a countable infinity of such orbits (an `explosion') and in which a countable infinity collapses to a unique orbit (an `implosion'). We also discover a rich variety of heteroclinic connections among fixed points and periodic orbits. The family we study was motivated by the search for travelling `structures' such as fronts and domain walls in the complex Ginzburg-Landau partial differential equation which models weakly nonlinear wave interactions near the onset of instability, and which leads to a special case of our more general unfolding.