A new procedure for analyzing the stochastic Hopf bifurcation of quasi-non-integrable-Hamiltonian systems is proposed. A quasi-non-integrable-Hamiltonian system is

An n-degree-of-freedom Hamiltonian system with r (1＜r＜n) integrals of motion which are in involution is called partially integrable Hamiltonian system. In the present paper, the exact stationary solutions of stochastically excited and dissipated partially integrable Hamiltonian systems are first reviewed. Then an equivalent non-linear system method for this class of systems in both nonresonant and resonant cases is developed. Three criteria are proposed to obtain the equivalent non-linear systems. The application and e!ectiveness of the method are illustrated by an example.

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegra-ble Hamiltonian system, the exact stationary solution is a functional of the Hamilto-nian and has the protYerty of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and binations of phase angles in resonant case with a (1≤α≤n-1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonin-tegrable Hamiltonian systems, which are further generalized to account for the modi-fication of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamilto-nian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.