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Eventually vanished solutions of a forced Liénard system

  • Yong-xin Zhang (张永新)Email author
Article

Abstract

In this paper, we aim to find eventually vanished solutions, a special class of bounded solutions which tend to 0 as t → ± ∞, to a Liénard system with a time-dependent force. Since it is not a Hamiltonian system with small perturbations, the well-known Melnikov method is not applicable to the determination of the existence of eventually vanished solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. Difficulties caused by the non-Hamiltonian form are overcome by applying the Schauder’s fixed point theorem. We show that the sequence of the periodic solutions has an accumulation giving an eventually vanished solution of the forced Liénard system.

Key words

eventually vanished bounded solution non-Hamiltonian accumulation 

Chinese Library Classification

O175.12 

2000 Mathematics Subject Classification

34A34 34C99 

References

  1. [1]
    Hahan, W. Stability of Motion, Springer-Verlag, Berlin and New York (1967)Google Scholar
  2. [2]
    Yoshizawa, T. Stability Theory by Liapunov’s Second Method, The mathematical Society of Japan, Takyo (1996)Google Scholar
  3. [3]
    Hale, J. K. Ordinary Differential Equations, 2nd Edition, Willey-Interscience, New York (1980)zbMATHGoogle Scholar
  4. [4]
    Buica, A., Gasull, A., and Yang, J. The third order Melnikov function of a quadratic center under quadratic perturbations. J. Math. Anal. Appl. 331(1), 443–454 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Champneys, A. and Lord, G. Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem. Physica D 102(1–2), 101–124 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Chow, S., Hale, J., and Mallet-Paret, J. An example of bifurcation to homoclinic orbits. J. Diff. Equ. 37(3), 351–371 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Dumortier, F., Li, C., and Zhang, Z. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Diff. Equ. 139(1), 146–193 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Li, C. and Rousseau, C. A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: the cusp of order 4. J. Diff. Equ. 79(1), 132–167 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Zhu, C. and Zhang, W. Computation of bifurcation manifolds of linearly independent homoclinic orbits. J. Diff. Equ. 245(7), 1975–1994 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Mawhin, J. and Ward, J. Periodic solutions of second order forced Liénard differential equations at resonance. Arch. Math. 41(2), 337–351 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Omari, P., Villari, G., and Zanolin, F. Periodic solutions of Liénard differential equations with one-sided growth restriction. J. Diff. Equ. 67(2), 278–293 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Franks, J. Generalizations of the Poincaré-Birkhoff theorem. Ann. Math. 128(1), 139–151 (1988)CrossRefMathSciNetGoogle Scholar
  13. [13]
    Jacobowitz, H. Periodic solutions of \( \ddot x \) + f(x, t) = 0 via Poincaré-Birkhoff theorem. J. Diff. Equ. 20(1), 37–52 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Andronov, A., Vitt, E., and Khaiken, S. Theory of Oscillators, Pergamon Press, Oxford (1966)zbMATHGoogle Scholar
  15. [15]
    Guckenheimer, J. and Holmes, P. Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York (1983)Google Scholar
  16. [16]
    Rabinowitz, P. H. Homoclinic orbits for a class of Hamiltonian systems. Proc. Roy. Soc. Edinburgh 114A(1), 33–38 (1990)MathSciNetGoogle Scholar
  17. [17]
    Ambrosetti, A. and Rabinowitz, P. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14(2), 349–381 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Chow, S. and Hale, J. Methods of Bifurcation Theory, Springer, New York (1982)zbMATHGoogle Scholar
  19. [19]
    Carrião, P. and Miyagaki, O. Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 230(1), 157–172 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Szulkin, A. and Zou, W. Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187(1), 25–41 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Izydorek, M. and Janczewska, J. Homoclinic solutions for a class of the second order Hamiltonian systems. J. Diff. Equ. 219(2), 375–389 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Izydorek, M. and Janczewska, J. Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 335(2), 1119–1127 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Tang, X. and Xiao, L. Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 351(2), 586–594 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    Tang, X. and Xiao, L. Homoclinic solutions for a class of second-order Hamiltonian systems. J. Math. Anal. Appl. 354(2), 539–549 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Zelati, V. C., Ekeland, I., and Séré, E. A variational approach to homoclinic orbits in Hamiltonian systems. Math. Ann. 288(1), 133–160 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    Sansone, G. and Conti, R. Nonlinear Differential Equations, Pergamon Press, New York (1964)Google Scholar

Copyright information

© Shanghai University and Springer Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of MathematicsSichuan UniversityChengduP. R. China

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