HOMOCLINIC TANGLES, PHASE-LOCKING, AND CHAOS IN A 2-FREQUENCY PERTURBATION OF DUFFINGS EQUATION

We study a two-frequency perturbation of Duffing's equation. When the perturbation is small, this system has a normally hyperbolic invariant torus which may be subjected to phase locking. Applying a version of Melnikov's method for multifrequency systems, we detect the occurrence of transverse intersection between the stable and unstable manifolds of the invariant torus. We show that if the invariant torus is not sub jected to phase locking, then such a transverse intersection yields ch aotic dynamics. When the invariant torus is subjected to phase locking , the situation is different. In this case, there exist two periodic o rbits which are created in a saddle-node bifurcation. Using another ve rsion of Melnikov's method for slowly varying oscillators, we also giv e conditions under which the stable and unstable manifolds of the peri odic orbits intersect transversely and hence chaotic dynamics may occu r. Our results reveal that when the invariant torus is subjected to ph ase locking, chaotic dynamics resulting from transverse intersection b etween its stable and unstable manifolds may be interrupted.