Stochastic stability of quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises

A procedure for determining the asymptotic Lyapunov stability with probability one of multi-degree-of-freedom (MDOF) quasi-integrable and resonant Hamiltonian systems under parametric excitations of combined Gaussian and Poisson white noises is proposed. For the case of resonance with α resonant relations, the averaged Ito^ stochastic differential equations (SDEs) for quasi-integrable and resonant Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises are first derived by using the stochastic averaging method. Then, the expression for the largest Lyapunov exponent is obtained by generalizing Khasminskii׳s procedure to the averaged Ito^ SDEs and using the property of the stochastic integro-differential equations (SIDEs). Finally, the stochastic stability of the original system is determined approximately by using the largest Lyapunov exponent. An example of two non-linear damping oscillators under parametric excitations of combined Gaussian and Poisson white noises is worked out to illustrate the application of the proposed procedure. The validity of the proposed procedure is verified by the good agreement between the analytical results and those from Monte Carlo simulation.