Global bifurcation and chaos in parametrically forced systems with one-one resonance

Abstract

Global bifurcations of a fourth-order Hamiltonian system with Z2 ⨭ Z2 symmetry are studied. The system represents normal-form equations that arise in a variety of problems which have one-one internal resonance and which are forced sinusoidally at the natural frequencies. Four qualitatively different types of global behaviours are shown to occur. Using a generalization of the Melnikov method, three different heteroclinic cycles are shown to break, generating Smale horseshoes and resulting in chaotic phenomena. The theoretical results are verified by numerical simulations. The main conclusion of the analysis is that chaotic phenomena are very common in this class of system