Many pulses heteroclinic orbits with a Melnikov method and chaotic dynamics of a parametrically and externally excited thin plateMinghui

The multi-pulse heteroclinic orbits with a Melnikov method and chaotic dynamics in a parametrically and externally excited thin plate are studied in this paper for the first time. The thin plate is subjected to transversal and in-plane excitations, simultaneously. The formulas of the thin plate are derived from the von Kármán equation and Galerkin’s method. The method of multiple scales is used to find the averaged equation. The theory of normal form, based on the averaged equation, is used to obtain the explicit expressions of normal form associated with a double zero and a pair of purely imaginary eigenvalues from the Maple program. Based on normal form obtained above, an extension of the Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics in a parametrically and externally excited thin plate. The global dynamics analysis indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equations for a parametrically and externally excited thin plate. These results show that the chaotic motions of the multi-pulse Shilnikov type can occur in a parametrically and externally excited thin plate. Numerical simulations are given to verify the analytical predictions. It is also found from the results of numerical simulation that the multipulse Shilnikov type orbits exist in a parametrically and externally excited thin plate.