中国物理B ›› 2008, Vol. 17 ›› Issue (2): 557-568.doi: 10.1088/1674-1056/17/2/034

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Chaotic motion of the dynamical system under both additive and multiplicative noise excitations

李秀春, 徐 伟, 李瑞红   

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China
  • 收稿日期:2006-10-31 修回日期:2007-09-07 出版日期:2008-02-20 发布日期:2008-02-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10472091 and 10332030).

Chaotic motion of the dynamical system under both additive and multiplicative noise excitations

Li Xiu-Chun(李秀春)†, Xu Wei(徐伟), and Li Rui-Hong(李瑞红)   

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China
  • Received:2006-10-31 Revised:2007-09-07 Online:2008-02-20 Published:2008-02-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant Nos 10472091 and 10332030).

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摘要： With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincar\'{e} map.

关键词: Melnikov theory, bounded noise, Lyapunov exponent, Poincar\'{e} map

Abstract: With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.

Key words: Melnikov theory, bounded noise, Lyapunov exponent, Poincaré map

中图分类号:  (Noise)

  • 05.40.Ca
05.45.Pq (Numerical simulations of chaotic systems)