Stochastic stability and bifurcation of quasi-Hamiltonian systems

The paper consists of two parts. In the first part the stochastic averaging of quasi-integrable-Hamiltonian systems with real noise excitations is introduced. The expression for the largest Lyapunov exponent of the square root of the Hamiltonian is formulated by generalizing the Khasminskii's procedure to the averaged systems, based on which the stochastic stability and bifurcation of the original systems are studied. In the second part, an n-degree-of-freedom quasi-non-integrable-Hamiltonian system is reduced to an Ito equation of one-dimensional averaged Hamiltonian by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. The necessary and sufficient conditions for the asymptotic stability in probability of the trivial solution and the condition for the Hopf bifurcation of the original systems are obtained approximately by examining the sample behaviors of the one-dimensional diffusion process of the square-root of averaged Hamiltonian and the averaged Hamiltonian, respectively, at the two boundaries.