STOCHASTIC AVERAGING OF QUASI-INTEGRABLE HAMILTONIAN-SYSTEMS

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree- of-freedom integrable Hamiltonian systems subject to lightly linear an d (or) nonlinear dampings and weakly external and (or) parametric exci tations of Gaussian white noises. According to the present method an n -dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n indepen dent integrals of motion can be constructed in nonresonant case. In a resonant case with alpha resonant relations, an (n + alpha)-dimensiona l averaged FPK equation governing the transition probability density o f n action variables and alpha combinations of phase angles can be obt ained. The procedures for obtaining the stationary solutions of the av eraged FPK equations for both resonant and nonresonant cases are prese nted. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.