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

A stochastic averaging method for predicting the response of quasi-partially integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. For the case with r(1<r<n) independent first integrals which are in involution, an r-dimensional averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for the transition probability density of r independent first integrals is derived from the stochastic integro-differential equations (SIDEs) of the original quasi-partially integrable and non-resonant Hamiltonian systems by using the stochastic jump-diffusion chain rule and the stochastic averaging theorem. An example is given to illustrate the applications of the proposed stochastic averaging method, and a combination of the finite difference method and the successive over-relaxation method is used to solve the reduced GFPK equation to obtain the stationary probability density of the system. The results are well verified by a Monte Carlo simulation.