Advertisement

Bifurcation and Chaos in Time-Delayed Piecewise Linear Dynamical System

  • M. Lakshmanan
  • D.V. Senthilkumar
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

The phenomena of bifurcations and chaos have been well studied in nonlinear dynamical systems without delay and described by nonlinear difference, differential, difference-differential, etc. equations. Routes to bifurcations, onset of chaos, nature of the chaotic attractors and their characterizations in such systems have all been analyzed extensively [1, 2]. However, such analyses have not been carried out in any greater detail in the case of nonlinear dynamical systems with delay even in the scalar systems. In this chapter we shall introduce a prototypical delay system, which is a piecewise linear one, in order to appreciate the nature of bifurcations and chaos phenomena underlying nonlinear time-delay systems, and to understand clearly the nature of transients and difficulties in numerical analysis as well as the frequent existence of hyperchaotic attractors with multiple positive Lyapunov exponents. The dynamics of other nonlinear time-delay systems will be taken up in the next chapter.

References

  1. 1.
    M. Lakshmanan, S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos and Patterns (Springer, Berlin, 2003)zbMATHGoogle Scholar
  2. 2.
    J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical System and Bifurcation of Vector Fields (Springer, Berlin, 1983)Google Scholar
  3. 3.
    H. Lu, Z. He, IEEE Trans. Circuits Syst. I 43, 700 (1996)CrossRefGoogle Scholar
  4. 4.
    P. Thangavel, K. Murali, M. Lakshmanan, Int. J. Bifurcat. Chaos 8, 2481 (1998)zbMATHGoogle Scholar
  5. 5.
    D.V. Senthilkumar, M. Lakshmanan, Int. J. Bifurcat. Chaos 15, 2895 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    K.H. Becker, M. Dorfler, Dynamical Systems and Fractals (Cambridge University Press, Cambridge, 1989)zbMATHCrossRefGoogle Scholar
  7. 7.
    D.V. Ramana Reddy, A. Sen, G.L. Johnston, Phys. Rev. Lett. 85, 3381 (2000)ADSCrossRefGoogle Scholar
  8. 8.
    M.G. Eral, S.H. Strogatz, Phys. Rev. E 67, 036204 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    S.A. Campbell, S. Ruan, J. Wei, Int. J. Bifurcat. Chaos 9, 1585 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    J.D. Farmer, Physica D 4, 366 (1982)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    W. Horbelt, J. Timmer, H.U. Voss, Phys. Lett. A 299, 513 (2002)ADSzbMATHCrossRefGoogle Scholar
  12. 12.
    H.U. Voss, Int. J. Bifurcat. Chaos 12, 1619 (2002)Google Scholar
  13. 13.
    Y. Pomeau, P. Manneville, J. Wei, Physica D 1, 219 (1980)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    H.G. Schuster, Deterministic Chaos (Springer, Weinheim, 1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Centre for Nonlinear Dynamics, Bharathidasan UniversityTiruchirapalliIndia
  2. 2.Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact ResearchPotsdamGermany

Personalised recommendations