Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system
Zhang Ying(张莹)a)†, Xu Wei(徐伟)a), Fang Tong(方同)b), and Xu Xu-Lin(徐旭林)c)
a Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China; b Vibration Research Centre, Northwestern Polytechnical University, Xi'an 710072, China; c Department of Automation, Nankai University, Tianjin 300071, China
Abstract In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer--van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.
Received: 16 October 2006
Revised: 16 November 2006
Accepted manuscript online:
(Fluctuation phenomena, random processes, noise, and Brownian motion)
Fund: Project supported by the Major Program of the National
Natural Science Foundation of China, China (Grant No~10332030), the National
Natural Science Foundation of China (Grant No~10472091), and the Graduate
Starting Seed Fund of Northwestern Polytechnical
Cite this article:
Zhang Ying(张莹), Xu Wei(徐伟), Fang Tong(方同), and Xu Xu-Lin(徐旭林) Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system 2007 Chinese Physics 16 1923
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