In the long course of the development of the modern theory of dynamical systems, ordinary differential equations have served as both a constant inspiration and a test ground. Periodically forced second order equations have been studied extensively in history. When an autonomous system with a saddle point and a homoclinic solution is subjected to periodic perturbations, the stable and unstable manifold of the perturbed saddle intersect transversally, creating a homoclinic tangle for the time-T map. Milnikov's method has been introduced for the purpose of detecting this homoclinic tangle in concrete systems of differential equations.

In this talk we follow a new route. Instead of studying the time-T map, about which our knowledge has been based mainly on geometric descriptions, we construct a Poincare section that mixes the original phase dimensions with time in the extended phase space and compute explicitly the return maps induced by the differential equations. It has turned out that, for dissipative equations, these return maps are rank one maps studied intensively by Young and myself in the last ten years. The conclusions of these studies, consequently, are applied in a rather natural fashion to the analysis of periodically forced second order equations with homoclinic solutions.

This talk is designed for a general audience, and the prerequisites are kept to a minimum.