Subharmonic Melnikov theory for degenerate resonance systems and its application

This article explores the subharmonic orbits of a four-dimensional degenerate resonance non-autonomous system. The well-known subharmonic Melnikov method is improved in this paper. Suppose the unperturbed system is a two-degree-of-freedom decoupled Hamiltonian system and possesses a family of periodic orbits. The important issue is the persistence of the periodic orbits after periodic perturbations. In order to solve this problem, we propose a four-dimensional subharmonic Melnikov function based on periodic transformations and Poincaré map. Then, a main theorem, which can be used to check the existence of the periodic orbits for the four-dimensional degenerate resonance system, is presented and proved by using the implicit function theorem. In order to verify the validity and applicability of the improved subharmonic Melnikov method, we apply it to investigate the subharmonic orbits of a honeycomb sandwich plate. The theoretical result shows that the subharmonic orbits of period 2 T can exist under certain conditions. Numerical simulations are carried out to verify the analytical predictions. It is found that the subharmonic orbits for the honeycomb sandwich plate do exist based on the results of numerical simulations.