Homoclinic orbits and chaos in three– and four–dimensional flows

We review recent work in which perturbative, geometric and topological arguments are used to prove the existence of countable sets of orbits connecting equilibria in ordinary differential equations. We first consider perturbations of a three–dimensional integrable system possessing a line of degenerate saddle points connected by a two–dimensional manifold of homoclinic loops. We show that this manifold splits to create transverse homoclinic orbits, and then appeal to geometrical and symbolic dynamic arguments to show that homoclinic bifurcations occur in which ‘simple’ connecting orbits are replaced by a countable infinity of such orbits. We discover a rich variety of connections among equilibria and periodic orbits, as well as more exotic sets, including Smale horseshoes. The second problem is a four–dimensional Hamiltonian system. Using symmetries and classical estimates, we again find countable sets of connecting orbits. There is no small parameter in this case, and the methods are non–perturbative.