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Center Manifold Analysis of the Delayed Lienard Equation

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Delay Differential Equations

Abstract

In this chapter, the authors show the existence of the Hopf bifurcation in the delayed Liénard equation. The criterion for the criticality of the Hopf bifurcation is established based on the reduction of the infinite-dimensional problem onto a twodimensional center manifold. Numerics based on DDE-Biftool are given to compare with the authors’ theoretical calculation. The Liénard type sunflower equation is discussed as an illustrative example based on our method.

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References

  1. N. MacDonald. Biological Delay Systems. Cambridge University Press, New York, 1989.

    Google Scholar 

  2. Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering, 191, 1993.

    Google Scholar 

  3. A. Beuter, J. Bélair, C. Labrie, and J. Bélair. Feedback and Delays in Neurological Diseases: A Modeling Study Using Gynamical Systems. Bulletin of Mathematical Biology, 55(3): 525–541, 1993.

    MATH  Google Scholar 

  4. H.Y. Hu. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Springer, New York, 2002.

    Google Scholar 

  5. G. Stépán. Modelling Nonlinear Regenerative Effects in Metal Cutting. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 359(1781):739–757, 2001.

    Article  MATH  Google Scholar 

  6. T. Kalmár-Nagy, G. Stépán, and F.C. Moon. Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations. Nonlinear Dynamics, 26(2):121–142, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Pieroux, T. Erneux, T. Luzyanina, and K. Engelborghs. Interacting Pairs of Periodic Solutions Lead to Tori in Lasers Subject to Delayed Feedback. Physical Review E, 63(3):36211, 2001.

    Article  Google Scholar 

  8. K. Pyragas. Continuous Control of Chaos by Self-controlling Feedback. Physics Letters A, 170(6):421–428, 1992.

    Article  Google Scholar 

  9. G. Stépán. Retarded Dynamical Systems: Stability and Characteristic Functions. Longman Scientific & Technical, Marlow, New York, 1989.

    Google Scholar 

  10. H.Y. Hu, E.H. Dowell, and L.N. Virgin. Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback. Nonlinear Dynamics, 15(4):311–327, 1998.

    Article  MATH  Google Scholar 

  11. A. Casal and M.M. Freedman. A Poincaré-Lindstedt Approach to Bifurcation Problems for Differential-delay Equations. IEEE Transactions on Automatic Control, 25(5):967–973, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  12. H.C. Morris. A Perturbative Approach to Periodic Solutions of Delay-differential Equations. IMA Journal of Applied Mathematics, 18(1):15–24, 2001.

    Article  Google Scholar 

  13. N. MacDonald. Harmonic Balance in Delay-differential Equations. Journal of Sound and Vibration, 186(4):649–656, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. Lehman and S. Weibel. Moving Averages for Periodic Delay Differential and Difference equations. Lecture Notes in Control and Information Science, 228:158–183.

    Google Scholar 

  15. B.D. Hassard, N.D. Kazarinoff, and Y.H. Wan. Theory and Applications of Hopf Bifurcation. London Mathematical Society Lecture Note Series, 41, 1981.

    Google Scholar 

  16. J.K. Hale. Theory of Functional Differential Equations. Applied Mathematical Sciences, vol. 3, 1977.

    Google Scholar 

  17. J.K. Hale. Nonlinear Oscillations in Equations with Time delay, in K. Hoppenstadt, Nonlinear Oscillations in Biology, (AMS,Providence, RI) 157-185, 1979.

    Google Scholar 

  18. W. Luk. Some Results Concerning the Boundedness of Solutions of Liénard Equations With Delay. SIAM Journal on Applied Mathematics, 30(4):768–774, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  19. B. Zhang. On the Retarded Liénard Equation. Proceedings of the American Mathematical Society, 115(3):779–785, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Zhang. Periodic Solutions of the Retarded Liénard Equation. Annali di Matematica Pura ed Applicata, 172(1):25–42, 1997.

    Article  MATH  Google Scholar 

  21. P. Omari and F. Zanolin. Periodic Solutions of Liénard Equations. Rendiconti del Seminario Matematico della Università di Padova, 72:203–230, 1984.

    MATH  MathSciNet  Google Scholar 

  22. G. Metzen. Existence of Periodic Solutions of Second Order Differential Equations with Delay. Proceedings of the American Mathematical Society, 103(3):765–772, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Acosta and M. Lizana. Hopf Bifurcation for the Equation x′′ (t) + f(x(t))x′ (t) + g(x(t − r)) = 0. Divulgaciones Matematicas, 13:35–43, 2005.

    MATH  MathSciNet  Google Scholar 

  24. J. Xu and Q.S. Lu. Hopf Bifurcation of Time-delay Liénard Equations. International Journal of Bifurcation and Chaos, 9(5):939–951, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  25. S.A. Campbell, J. Bélair, T. Ohira, and J. Milton. Limit cycles, Tori, and Complex Dynamics in a Second-order Differential Equation with Delayed Negative Feedback. Journal of Dynamics and Differential Equations, 7(1):213–236, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  26. F. Colonius and F. Box. Optimal Periodic Control of Retarded Liénard Equation. Distributed Parameter Systems, Proceedings of Science International Conference, Vorau/Austria, 1984.

    Google Scholar 

  27. K. Engelborghs, T. Luzyanina, and D. Roose. Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL. ACM Transactions on Mathematical Software, 28(1):1–21, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  28. K. Engelborghs, T. Luzyanina, and G. Samaey. DDE-BIFTOOL v. 2.00: A Matlab Package for Bifurcation Analysis of Delay Differential Equations. Report TW, 330, 2001.

    Google Scholar 

  29. D. Israelson and A. Johnson. Theory of Circumnutations in Helianthus Annus. Physiology Plant, 20:957–976, 1967.

    Article  Google Scholar 

  30. A.S. Somolinos. Periodic Solutions of the Sunflower Equation: x + (a/r)x + (b/r)sin x(t − r) = 0. Q. Appl. Math. (Quarterly of Applied Mathematics) 35:465–478, 1978.

    MATH  MathSciNet  Google Scholar 

  31. L.P. Liu and T. Kalmár-Nagy. High Dimensional Harmonic Balance Analysis for Second-Order Delay-Differential Equations. Journal of Vibration and Control, in press.

    Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, Texas A&M University, 609C H. R. Bright Building, College Station, TX, 77843, USA

    Tamás Kalmár-Nagy

Authors
  1. Siming Zhao
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  2. Tamás Kalmár-Nagy
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Correspondence to Tamás Kalmár-Nagy .

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© 2009 Springer-Verlag US

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Zhao, S., Kalmár-Nagy, T. (2009). Center Manifold Analysis of the Delayed Lienard Equation. In: Delay Differential Equations. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-85595-0_7

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  • DOI: https://doi.org/10.1007/978-0-387-85595-0_7

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  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-0-387-85594-3

  • Online ISBN: 978-0-387-85595-0

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