A high-dimensional Melnikov method is developed to study bifurcations of multiple periodic solutions of nonlinear dynamical systems with parameters， and applied to study of complex nonlinear dynamical behaviors such as multiple periodic motions of honeycomb sandwich plates with negative Poisson''s ratio. By constructing the curvilinear coordinates and Poincaré map， the Melnikov function suitable for a four-dimensional nonlinear dynamical system with parameters is developed. The decision theorems on existence and number of multiple periodic solutions are obtained. Based on the theoretical results， the multiple periodic motions of honeycomb sandwich plate with negative Poisson''s ratio under in-plane and transverse excitations are investigated. The existence， number and parameter control conditions of periodic orbits are derived. The influence of transverse excitation on the system''s dynamical behaviors is discussed. Under certain parameter conditions， there exist at most four periodic orbits， and the phase portrait configurations are given by numerical simulations to verify the theoretical results.