中国物理B ›› 2006, Vol. 15 ›› Issue (6): 1231-1238.doi: 10.1088/1009-1963/15/6/017

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Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation

徐伟1, 李伟1, 方同1, 马少娟2   

  1. (1)Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China; (2)Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China；Department of Information & Computation Sciences, the Second Northwest University for Nationalities, Yinchuan, 750021,China
  • 收稿日期:2005-06-23 修回日期:2006-03-16 出版日期:2006-06-20 发布日期:2006-06-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grants Nos 10472091 and 10332030).

Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation

Ma Shao-Juan (马少娟)ab, Xu Wei (徐伟)a, Li Wei (李伟)a, Fang Tong (方同)a   

  1. a Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China; b Department of Information & Computation Sciences, the Second Northwest University for Nationalities, Yinchuan 750021, China
  • Received:2005-06-23 Revised:2006-03-16 Online:2006-06-20 Published:2006-06-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grants Nos 10472091 and 10332030).

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摘要： 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.

关键词: stochastic Duffing--van der Pol system, Chebyshev polynomial approximation, stochastic period-doubling bifurcation, stochastic chaos

Abstract: 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.

Key words: stochastic Duffing--van der Pol system, Chebyshev polynomial approximation, stochastic period-doubling bifurcation, stochastic chaos

中图分类号:  (Synchronization; coupled oscillators)

  • 05.45.Xt
02.30.Oz (Bifurcation theory) 02.50.Fz (Stochastic analysis) 05.45.Ra (Coupled map lattices)