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

A stochastic averaging method for approximately predicting the responses of multi-degree-of-freedom (MDOF) quasi-non-integrable Hamiltonian systems (non-integrable Hamiltonian systems with lightly linear and (or) nonlinear dampings) to weakly external and (or) parametric excitations of Gaussian and Poisson white noises is proposed. By means of the stochastic integro-differential equations (SIDE) and stochastic jump-diffusion chain rule, a one-dimensional averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for the transition probability density of the Hamiltonian is derived to obtain the probability density and statistics of the stationary responses through solving the GFPK equation using perturbation technique. An example is given to illustrate the application of the proposed method. It is shown that theoretical results agree well with those from Monte Carlo simulation.