Period-doubling bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation

Chebyshev polynomial approximation is applied to the period-doubling bifurcation problem of a stochastic van der Pol system with bounded random parameters and subjected to harmonic excitations. Firstly, the stochastic system is reduced to its equivalent deterministic one, through which the response of the stochastic system can be obtained by numerical methods. Nonlinear dynamical behavior related to various forms of stochastic period-doubling bifurcation in the stochastic system is explored. Numerical simulations show that similar to their counterpart in deterministic nonlinear system, various forms of period-doubling bifurcation may occur in the stochastic van der Pol system, but with some modified features. Numerical results also show that Chebyshev polynomial approximation can provide an effective approach to dynamical problems in stochastic nonlinear systems.