中国物理B ›› 2007, Vol. 16 ›› Issue (7): 1923-1933.doi: 10.1088/1009-1963/16/7/020

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Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system

张莹1, 徐伟1, 徐旭林2, 方同3   

  1. (1)Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China; (2)Department of Automation, Nankai University, Tianjin 300071, China; (3)Vibration Research Centre, Northwestern Polytechnical University, Xi'an 710072, China
  • 收稿日期:2006-10-16 修回日期:2006-11-16 出版日期:2007-07-20 发布日期:2007-07-04
  • 基金资助:
    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

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)   

  1. 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
  • Received:2006-10-16 Revised:2006-11-16 Online:2007-07-20 Published:2007-07-04
  • Supported by:
    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

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摘要： 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.

关键词: Chebyshev polynomial approximation, stochastic Bonhoeffer--van der Pol system, \\ \hspace*{1.9cm} stochastic period-doubling bifurcation, bounded random parameter

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.

Key words: Chebyshev polynomial approximation, stochastic Bonhoeffer--van der Pol system, stochastic period-doubling bifurcation, bounded random parameter

中图分类号:  (Nonlinear dynamics and chaos)

  • 05.45.-a
02.30.Oz (Bifurcation theory) 02.50.Ey (Stochastic processes) 02.60.Cb (Numerical simulation; solution of equations) 05.40.-a (Fluctuation phenomena, random processes, noise, and Brownian motion)