ORBITS HOMOCLINIC TO RESONANCES, WITH AN APPLICATION TO CHAOS IN A MODEL OF THE FORCED AND DAMPED SINE-GORDON EQUATION

In this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two-degree-of-freedom Hamiltonian systems. Our methods are fundamentally different than other global perturbation methods (e.g. standard Melnikov theory) in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation. Our methods combine the higher dimensional Melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds. We apply OUT methods to a modified model of the forced and damped sine-Gordon equation developed by Bishop et al. We give explicit conditions (in terms of the system parameters) for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle-focus type; chaotic dynamics follow from a theorem of Silnikov. This provides a mechanism for chaotic dynamics geometrically similar to that observed by Bishop et al.; namely, a random "jumping" between two spatially dependent states with an intermediate passage through a spatially independent state. However, in order for this type of Silnikov dynamics to exist we require a different, and unphysical, type of damping compared to that used by Bishop et al.