Abstract
Ann-degree-of-freedom quasi-non-integrable-Hamiltonian system is first reduced to anItôequation of one-dimensional averaged Hamiltonian by using the stochastic averaging method developed by the first author and his coworkers. The necessary and sufficient conditions for the asymptotic stability in probability of the trivial solution of the quasi-non-integrable-Hamiltonian system are then obtained approximately by examining the sample behaviors of the one-dimensional diffusion process of the square-root of averaged Hamiltonian at the two boundaries. A system of linearly and non-linearly coupled two non-linearly damped oscillators subject to parametric excitations of Gaussian white noises is employed as an example to illustrate the procedure, and the effects of non-linear damping and non-linear coupling on the stability are analyzed in detail.
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