Stochastic Dynamics Of Quasi Integrable Hamiltonian Systems With Fractional Derivative Damping Under Combined Harmonic And White (Wide Band) Noise Excitations

The stochastic response, first passage failure and stochastic stability of quasi integrable Hamiltonian systems with fractional derivative damping subjected to combined harmonic and Gauss white noise excitations or combined harmonic and wide band noise excitations are investigated. First, the stochastic averaged equations for quasi integrable Hamiltonian systems with fractional derivative damping under combined harmonic and Gauss white (wide band) noise excitations are derived by using the generalized harmonic functions. It is pointed out that the form and dimension of averaged Ito equations depend upon the resonance of systems. The drift and diffusion coefficients of the averaged Ito equation are given. Then, the Fokker-Planck-Kolmogorov ( FPK) equations governing the probability density of response processes, the backward Kolmogorov governing the conditional reliability function and the Pontryagin equation governing the mean first passage time are established. The response statistics, the conditional reliability function, the conditional probability density and mean of first passage time of the system are obtained by solving these equations, respectively. The effects of fractional order on the stochastic response and reliability are discussed via a series of examples, respectively. In addition, the first passage failure of two nonlinearly coupled Duffing oscillators subjected to combined harmonic and Gaussian white noises excitations is investigated. Finally, the asymptotic Lyapunov stability with probability one of combined harmonic and white (wide band) noise excited quasi integrable Hamiltonian systems with fractional derivative damping in the cases of non-resonance and resonance is studied by using the largest Lyapunov exponent and the effect of fractional order on the asymptotic stability of a single-degree-of-freedom system is investigated.