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

We review recent work in which perturbative, geometric and topological argu ments are used to prove the existence of countable sets of orbits connectin g equilibria in ordinary differential equations. We first consider perturbations of a three-dimensional integrable system po ssessing a line of degenerate saddle points connected by a two-dimensional manifold of homoclinic loops. We show that this manifold splits to create t ransverse homoclinic orbits, and then appeal to geometrical and symbolic dy namic 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 orbit s, as well as more exotic sets, including Smale horseshoes. The second problem is a four-dimensional Hamiltonian system. Using symmetri es and classical estimates, we again find countable sets of connecting orbi ts. There is no small parameter in this case, and the methods are non-pertu rbative.