Global bifurcations in parametrically excited systems with zero-to-one internal resonance

In this work we study the existence of Silnikov homoclinic orbits in the averaged equations representing the modal interactions between two modes with zero-to-one internal resonance. The fast mode is parametrically excited near its resonance frequency by a periodic forcing. The slow mode is coupled to the fast mode when the amplitude of the fast mode reaches a critical value so that the equilibrium of the slow mode loses stability. Using the analytical solutions of an unperturbed integrable Hamiltonian system, we evaluate a generalized Melnikov function which measures the separation of the stable and the unstable manifolds of an annulus containing the resonance band of the fast mode. This Melnikov function is used together with the information of the resonances of the fast mode to predict the region of physical parameters for the existence of Silnikov homoclinic orbits.