Some Recent Advances in Theory of Stochastically Excited Dissipative Hamiltonian Systems

Abstract

Some recent advances in the theory of stochastically excited dissipative Hamiltonian systems made by the author and his co-workers are summarized. It is shown that the structure of the solution and the energy partition among various degrees of freedom of a stochastically excited dissipative Hamiltonian system depend upon the integrability and resonance of the Hamiltonian system modified by the Wong-Zakai correction terms. Three procedures, i. e., one for obtaining exact stationary solution, equivalent nonlinear system method and stochastic averaging method, for predicting the response of stochastically excited dissipative Hamiltonian systems are presented. It is pointed out that all presently available exact stationary solutions of nonlinear stochastic systems can be obtained by the present procedure as special cases and that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are included in the present stochastic averaging of quasi-Hamiltonian systems as two special cases.