The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing-van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing-van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.
In this paper, the effect of every parameter （including p, q, r, λ, τ） on the mean first-passage time （MFPT） is investigated in an asymmetric bistable system driven by colour-correlated noise. The expression of MFPT has been obtained by applying the steepest-descent approximation. Numerical results show that （1） the intensity of multiplicative noise p and the intensity of additive noise q play different roles in the MFPT of the system, （2） suppression appears on the curve of the MFPT with small λ （e.g. λ 〈 0.5） but there is a peak on the curve of the MFPT when λ is big （e.g. λ 〉 0.5）, and （3） with different values of r （e.g. r = 0.1, 0.5, 1.5）, the effort of τ on the MFPT is diverse.
