Singularity Structure and Chaotic Dynamics of the Parametrically Driven Duffing Oscillator

  • S. Parthasarathy
Conference paper
Part of the Research Reports in Physics book series (RESREPORTS)


The nonintegrability aspects of the generalised parametrically driven Duffing oscillator is reviewed by investigating analytically the singularity structure exhibited by the solution of the system in the complex time domain. The simultaneous contraction and rotation of this singularity pattern in the z = t4 lnt plane, as the control parameter is varied, results in complicated clustering of singularities in the t-plane is pointed out. The corresponding chaotic dynamics of the system is studied numerically.


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  1. [1]
    M. Tabor and J. Weiss, Phys. Rev. A24 (1981) 2157.ADSGoogle Scholar
  2. [2]
    T. Bountis, H. Segur and M. Vivaldi, Phys. Rev. A25 (1982) 1257.MathSciNetADSGoogle Scholar
  3. [3]
    Y. F. Chang, J. M. Greene, M. Tabor and J. Weiss, Physica 8D (1983) 183.MathSciNetADSGoogle Scholar
  4. [4]
    T. Bountism, V. Papageorgiou and M. Bier, Physica 24D (1987) 292.ADSGoogle Scholar
  5. [5]
    J. D. Fournier, G. Levine and M. Tabor, J. Phys. A21 (1988) 33.MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    G. Levine and M. Tabor, Physica 33D (1988) 189.MathSciNetMATHGoogle Scholar
  7. [7]
    M. Tabor, Pramana J. Phys. 33 (1989) 315.Google Scholar
  8. [8]
    M. J. Ablowitz, A. Ramani and H. Segur, J. Math. Phys. 21 (1980) 715.Google Scholar
  9. [9]
    M. Lakshmanan and R. Sahadevan, Phys. Rev. A31 (1985) 861.MathSciNetADSGoogle Scholar
  10. [10]
    M. Lakshmanan and R. Sahadevan, “Painlevé Analysis, Lie Symmetries and Integrability of Coupled Nonlinear Oscillators of Polynomial Type,” Phys. Rep., to appear (1990).Google Scholar
  11. [11]
    A. Ramani, B. Grammaticos and T. Bountis, Phys. Rep. 180 (1989) 160.MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    V. K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 1.Google Scholar
  13. [13]
    G. Duffing, Erzwnungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung ( Vieweg, Braundschweig, 1918 ).Google Scholar
  14. [14]
    J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields(Springer, Berlin,1983).Google Scholar
  15. [15]
    F. C. Moon, Chaotic Vibrations ( John Wiley, New York, 1987 ).MATHGoogle Scholar
  16. [16]
    R. A. Mahaffey, Phys. Fluids 19 (1976) 1387.ADSCrossRefGoogle Scholar
  17. [17]
    Y. H. Kao, J. C. Huang and Y. S. Gou, Phys. Rev. A35 (1987)5228.ADSGoogle Scholar
  18. [18]
    H. T. Davis, Introduction to Nonlinear Differential and Integral Equations ( Dover, New York, 1962 ).MATHGoogle Scholar
  19. [19]
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions ( Dover, New York, 1965 ).Google Scholar
  20. [20]
    S. Parthasarathy and M. Lakshmanan, J. Sound & Vibration 137 (1990) 523.MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • S. Parthasarathy
    • 1
  1. 1.Department of PhysicsBharathidasan UniversityTiruchirapalliIndia

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