First-passage problem for strong nonlinear stochastic dynamical systems

In this paper, the stochastic averaging method for stochastic and dissipative quasi-non-integrable Hamiltonian system is applied to study the first-passage problem of coupled Duffing–van der Pol system subject to Gaussian white noise excitation. First, a Backward Kolmogorov equation for the conditional reliability function and a Generalized Pontryagin equation for the conditional moment of the first-passage time are established. Then according to the classification of the boundary conditions and initial conditions of these two kinds of equations, the reliability of system response under external excitation and parametric excitation are analyzed, respectively, in detail. At last, numerical results for reliability function, the probability of first-passage time and mean first-passage time of system are given in virtue of figures.