Periodically Forced Double Homoclinic Loops to a Dissipative Saddle

In this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point. We prove the existence of many complicated dynamical objects for given equations, ranging from attractive quasi-periodic torus, to Newhouse sinks and Hénon-like attractors, and to rank one attractors with SRB measures and full stochastic behavior. This theory enables us to apply rigorously many profound dynamics theories on non-uniformly hyperbolic maps developed in the last forty years, including the Newhouse theory, the theory of SRB measures, the theory of Hénon-like attractors and the theory of rank one attractors, to the analysis of the strange attractors in a periodically perturbed Duffing equation.