HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

Applied Mathematics and Mechanics (English Edition) ›› 1992, Vol. 13 ›› Issue (3): 217-226.

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

张伟¹, 霍拳忠¹, 李骊²

1. Tianjin University, Tianjin;
2. Peking Polytechnic University, Beijing

收稿日期:1991-01-21 出版日期:1992-03-18 发布日期:1992-03-18

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

Zhang Wei¹, Huo Quan-zhong¹, Li Li²

1. Tianjin University, Tianjin;
2. Peking Polytechnic University, Beijing

Received:1991-01-21 Online:1992-03-18 Published:1992-03-18

摘要/Abstract

摘要： Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov’s method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.

关键词: heteroclinic orbit bifurcations, subharmonic bifurcations, chaotic motions, parametric excitation, Melnikov’s method

Abstract: Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov’s method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.

Key words: heteroclinic orbit bifurcations, subharmonic bifurcations, chaotic motions, parametric excitation, Melnikov’s method

张伟;霍拳忠;李骊. HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR[J]. Applied Mathematics and Mechanics (English Edition), 1992, 13(3): 217-226.

Zhang Wei;Huo Quan-zhong;Li Li. HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR[J]. Applied Mathematics and Mechanics (English Edition), 1992, 13(3): 217-226.

参考文献

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[5] Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.
[6] Greenspan, B. D., and P. J. Holems, Homoclinic orbits, subharmonics and global bifurcations in forced oscillations, Nonlinear Dynamics and Turbulence, G. Barenblatt, G. Ioose, and D. D. Joseph(eds), Pitman, London, (1983), 172-214.
[7] Melnikov, V. K., On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57.
[8] Holmes, P. J., Averaging and chaotic motions in forced oscillations, SIAM J. Appl. Math., 38(1980), 65-80.
[9] Hale, J. K., Ordinary Differential Equations, 2nd Edition, Kreiger Publ. Co. (1980).
[10] Hale, J. K. and X.-B. Lin, Heteroclinic orbits for retarded functional differential equation,J. Diff. Eqs., 65 (1986), 175-202.
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[13] Brunsden, V., J. Cortell and P. J. Holmes, Power spectra of chaotic vibrations of a buckled beam, J. Sound Vib., 130, 1(1989), 1-25.

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

HETEROCLINIC ORBIT AND SUBHARMONIC BIFURCATIONS AND CHAOS OF NONLINEAR OSCILLATOR

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