Lyapunov function method for analyzing stability of quasi-Hamiltonian systems under combined Gaussian and Poisson white noise excitations

The asymptotic Lyapunov stability with probability one of multi-degree-of-freedom quasi-Hamiltonian systems subject to parametric excitations of combined Gaussian and Poisson white noises is studied by using Lyapunov function method. According to the integrability and resonance, quasi-Hamiltonian systems can be divided into five classes, namely quasi-non-integrable, quasi-completely integrable and non-resonant, quasi-completely integrable and resonant, quasi-partially integrable and non-resonant, and quasi-partially integrable and resonant. Lyapunov functions for these five classes of systems are constructed. The derivatives for these Lyapunov functions with respect to time are obtained by using the stochastic averaging method. The approximately sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian system under parametric excitations of combined Gaussian and Poisson white noises is determined based on a theorem due to Khasminskii. Four examples are given to illustrate the application and efficiency of the proposed method. And the results are compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent method.