Stochastic averaging of quasi partially integrable and resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for quasi partially integrable and resonant Hamiltonian systems excited by the combined Gaussian and Poisson white noises is proposed. The averaged stochastic integro-differential equations (SIDEs) and generalized Fokker-Planck-Kolmogorov (GFPK) equation for the quasi partially integrable Hamiltonian system with r (1 < r < n) independent first integrals in involution and beta (1 < beta < r - 2) resonant relations are derived. The dimension of averaged SIDEs and the GFPK equation is equal to the number of the independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. A 4-degree-of-freedom (DOF) quasi partially integrable and resonant Hamiltonian system under combined Gaussian and Poisson white noise excitations are calculated as an example to illustrate the application of the proposed method. Different resonant cases and non-resonant case are worked out for the purpose of comparison. The Monte Carlo simulations are also carried out to verify the effectiveness and accuracy of the proposed stochastic averaging method.