中国物理B ›› 2008, Vol. 17 ›› Issue (11): 4123-4128.doi: 10.1088/1674-1056/17/11/027

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High-codimensional static bifurcations of strongly nonlinear oscillator

张琪昌, 王 炜, 刘富豪   

  1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China
  • 收稿日期:2008-05-16 修回日期:2008-06-03 出版日期:2008-11-20 发布日期:2008-11-20
  • 基金资助:
    Project supported by the National Natural Science Foundation of China (Grant No 10872141) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No 20060056005).

High-codimensional static bifurcations of strongly nonlinear oscillator

Zhang Qi-Chang (张琪昌), Wang Wei (王 炜), Liu Fu-Hao (刘富豪)   

  1. Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China; State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China
  • Received:2008-05-16 Revised:2008-06-03 Online:2008-11-20 Published:2008-11-20
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Grant No 10872141) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No 20060056005).

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摘要： The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing--van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with $Z_{2}$ symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.

关键词: bifurcation, strongly nonlinear, normal form, singularity theory

Abstract: The static bifurcation of the parametrically excited strongly nonlinear oscillator is studied. We consider the averaged equations of a system subject to Duffing--van der Pol and quintic strong nonlinearity by introducing the undetermined fundamental frequency into the computation in the complex normal form. To discuss the static bifurcation, the bifurcation problem is described as a 3-codimensional unfolding with $Z_{2}$ symmetry on the basis of singularity theory. The transition set and bifurcation diagrams for the singularity are presented, while the stability of the zero solution is studied by using the eigenvalues in various parameter regions.

Key words: bifurcation, strongly nonlinear, normal form, singularity theory

中图分类号:  (Nonlinear dynamics and chaos)

  • 05.45.-a
02.40.Xx (Singularity theory)