CHAOTIC MOTIONS NEAR HOMOCLINIC MANIFOLDS AND RESONANT TORI IN QUASI-PERIODIC PERTURBATIONS OF PLANAR HAMILTONIAN-SYSTEMS

We study chaotic dynamics of nonlinear oscillators with the form of a two-frequency quasiperiodic perturbation of a planar Hamiltonian syste m possessing a homoclinic orbit whose interior contains a one-paramete r family of periodic orbits. In the extended phase space the unperturb ed system has a three-dimensional homoclinic manifold and a one-parame ter family of invariant 3-tori. Using Melnikov's technique and the sec ond-order averaging method, we show that chaotic motions may exist nea r the unperturbed homoclinic manifold and the unperturbed resonant tor i. These chaotic motions result from transverse intersection between t he stable and unstable manifolds of normally hyperbolic invariant tori , and are characterized by a generalization of the Bernoulli shift. We also give an example for the quasiperiodically forced Duffing oscilla tor and demonstrate the existence of these chaotic motions by numerica l simulation.