Acta Mechanica Sinica

, Volume 11, Issue 4, pp 357–372 | Cite as

Global bifurcations and chaos in a Van der Pol-Duffing-Mathieu system with three-well potential oscillator

  • Chen Yushu
  • Xu Jian


Semi-analytical and semi-numerical method is used to investigate the global bifurcations and chaos in the nonlinear system of a Van der Pol-Duffing-Mathieu oscillator. Semi-analytical and semi-numerical method means that the autonomous system, called Van der Pol-Duffing system, is analytically studied to draw all global bifurcations diagrams in parameter space. These diagrams are called basic bifurcation diagrams. Then fixing parameter in every space and taking parametrically excited amplitude as a bifurcation parameter, we can observe the evolution from a basic bifurcation diagram to chaotic pattern by numerical methods.

Key Words

semi-analytical and semi-numerical method global bifurcations chaos van del Pol-Duffing-Mathieu system 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Holmes P, Moon F. Strange attractors and chaos in nonlinear mechanics.ASME J Appl Mech, 1983, 50: 1021–1032MathSciNetCrossRefGoogle Scholar
  2. [2]
    Zavodeney LD, Nayfeh AH, Sanchez NE. Bifurcations and chaos in parametrically excited single-degree-of-freedom systems.J Non Dyn, 1990, 1: 1–21CrossRefGoogle Scholar
  3. [3]
    Chen Yushu, Xu Jian. Periodic response and bifurcation theory of nonlinear Hill system.J Non Dyn Eng, 1993, 1: 1–14Google Scholar
  4. [4]
    Ueda Y. Randomly transational phenomena in the system governed by Duffing's equation.J Stat Phys, 1979, 2: 181–197MathSciNetCrossRefGoogle Scholar
  5. [5]
    Soliman MS, Thompson JMT. Intergrity measures quantifying the erosion of smooth and fractal basins of attraction.J Sound Vib, 1989, 135: 453–475MathSciNetCrossRefGoogle Scholar
  6. [6]
    Guckenheimer, Holmes J. Nonlinear Oscillation, Dynamical System and Bifurcation of Vector Fields. New York: Springer-Verlag, 1983Google Scholar
  7. [7]
    Bolotin VV. The Dynamic Stability of Elastic Systems. Holden-Day, 1964Google Scholar
  8. [8]
    Chen YS. Bifurcation and Chaos Theory of Nonlinear Vibration Systems. Beijing: High-Education Press, 1993Google Scholar
  9. [9]
    Levin RW, Pompe B, Wilke C, Koch BP. Experiments on periodic and chaotic motions of a parametrically forced pendulm.Phys D, 1985, 16: 371–384MathSciNetCrossRefGoogle Scholar
  10. [10]
    Holmes PJ, Rand D. Phase portrait and bifurcations of the nonlinear oscillator:x″+(α+γx 2)x′+βxx 3=0.Int J Nonlinear Mech, 1980, 15: 449–458zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Li CZ, Rousseau C. Codimension 2 symmetric homoclinic bifurcations and application to 1∶2 resonace.Can J Math, 1990, XL: 191–212MathSciNetGoogle Scholar
  12. [12]
    Takens F. Forced oscillation and bifurcations.Comm Math Inst Rijksuniversiteit Utrecht, 1974, 3: 1–59MathSciNetGoogle Scholar
  13. [13]
    Wiggins S. Global bifurcation and chaos applicated method. Springer-Verlag, 1988Google Scholar
  14. [14]
    Feigenbaum MJ. The universal metric properties of nonlinear transform.J Stat Phys, 1979, 21: 669–706zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Chinese Society of Theoretical and Applied Mechanics 1995

Authors and Affiliations

  • Chen Yushu
    • 1
  • Xu Jian
    • 1
  1. 1.Department of MechanicsTianjin UniversityTianjinChina

Personalised recommendations