Stochastic averaging of quasi-integrable Hamiltonian systems under combined harmonic and white noise excitations

Abstract

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems to combined harmonic and white noise excitations. According to the proposed method, an n+ α+ β-dimensional averaged Fokker–Planck–Kolmogorov (FPK) equation governing the transition probability density of n action variables or independent integrals of motion, α combinations of angle variables and β combinations of angle variables and excitation phase angles can be constructed when the associated Hamiltonian system has α internal resonant relations and the system and harmonic excitations have β external resonant relations. The averaged FPK equation is solved by using the combination of the finite difference method and the successive over relaxation method. Two coupled Duffing–van der Pol oscillators under combined harmonic and white noise excitations is taken as an example to illustrate the application of the proposed procedure and the stochastic jump and its bifurcation as the system parameters change are examined.