Bifurcations of Subharmonic Solutions of Four-dimensional Systems

Abstract

In this paper, the bifurcation of subharmonic orbits for a four-dimensional system is considered. Suppose the unperturbed system has a family of periodic orbits. The problem addressed here is the determination of sufficient conditions for some of the periodic orbits to generate subharmonic orbits after periodic perturbations. Based on periodic transformations and Poincare map we can obtain the bifurcation function which can be recognized as subharmonic Melnikov function. The method succeeds in establishing the existence of subharmonics in perturbed Hamiltonian systems as well as in discussing their bifurcations. The subharmonic Melnikov method is directly applied to investigate the subharmonic orbits for a simply supported rectangular thin plate under combined parametric and external excitations for the first time. Numerical simulations are finished to verify the analytical predictions.