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Homoclinic Solutions for a Forced Liénard Type System

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In this paper, we find a special class of homoclinic solutions which tend to 0 as t → ±∞, for a Liénard type system with a time-dependent force. Since it is not a small perturbation of a Hamiltonian system, we cannot employ the well-known Melnikov method to determine the existence of homoclinic solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. We prove that the sequence of those periodic solutions has an accumulation which gives an homoclinic solution of the forced Liénard type system.

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References

  1. Ambrosetti A., Rabinowitz P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andronov A., Vitt E., Khaiken S.: Teory of Oscillators. Pergamon Press, Oxford (1966)

    Google Scholar 

  3. Buica A., Gasull A., Yang J.: The third order Melnikov function of a quadratic center under quadratic perturbations. J. Math. Anal. Appl. 331, 443–454 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carrião P., Miyagaki O.: Existence of homoclinic solutions for a class of time-dependent Hamiltonian systems. J. Math. Anal. Appl. 230, 157–172 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Champneys A., Lord G.: Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem. Phys D: Nonlinear Phenom 102, 101–124 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chow S., Hale J., Mallet-Paret J.: An example of bifurcation to homoclinic orbits. J. Differ. Equ. 37, 351–371 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chow S., Hale J.: Methods of Bifurcation Theory. Springer, New York (1982)

    MATH  Google Scholar 

  8. Dumortier F., Li C., Zhang Z.: Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differ. Equ. 139, 146–193 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Franks J.: Generalizations of the Poincaré-Birkhoff theorem. Ann. Math. 128, 139–151 (1988)

    Article  MathSciNet  Google Scholar 

  10. Guckenheimer J., Holmes P.: Nonlinear Oscillation, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Google Scholar 

  11. Hahan W.: Stability of Motion. Springer-Verlag, Berlin (1967)

    Google Scholar 

  12. Hale J.K.: Ordinary Differential Equations, 2nd edn. Wiley-Interscience, New York (1980)

    MATH  Google Scholar 

  13. Izydorek M., Janczewska J.: Homoclinic solutions for a class of the second order Hamiltonian systems. J. Differ. Equ. 219, 375–389 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Izydorek M., Janczewska J.: Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 335, 1119–1127 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jacobowitz H.: Periodic solutions of ẍ + f(x, t) = 0 via Poincaré-Birkhoff theorem. J. Differ. Equ. 20, 37–52 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li C., 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. Differ. Equ. 79, 132–167 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  17. Massera J.L.: On Liapunoff’s conditions of stability. Ann. Math. 50, 705–721 (1949)

    Article  MathSciNet  Google Scholar 

  18. Mawhin J., Ward J.: Periodic solutions of second order forced Liénard differential equations at resonance. Arch. Math. 41, 337–351 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Omari P., Villari G., Zanolin F.: Periodic solutions of Liénard differential equations with one-sided growth restriction. J. Differ. Equ. 67, 278–293 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  20. Rabinowitz P.: Homoclinic orbits for a class of Hamiltonian systems. Proc. Roy. Soc. Edinburgh 114, 33–38 (1990)

    MATH  MathSciNet  Google Scholar 

  21. Reuter G.: Boundedness theorems for nonlinear differential equations of the second order. J. Lond. Math. Soc. 27, 48–58 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  22. Szulkin A., Zou W.: Homoclinic orbits for asymptotically linear Hamiltonian systems. J. Funct. Anal. 187, 25–41 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tang X., Xiao L.: Homoclinic solutions for nonautonomous second-order Hamiltonian systems with a coercive potential. J. Math. Anal. Appl. 351, 586–594 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tang X., Xiao L.: Homoclinic solutions for a class of second-order Hamiltonian systems. J. Math. Anal. Appl. 354, 539–549 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Yoshizawa T.: Stability Theory by Liapunov’s Second Method. The Math. Soc. of Japan, Tokyo (1996)

    Google Scholar 

  26. Zhu C., Zhang W.: Computation of bifurcation manifolds of linearly independent homoclinic orbits. J. Differ. Equ. 245, 1975–1994 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Leshan Normal University, 614004, Leshan, Sichuan, People’s Republic of China

    Yongxin Zhang

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  1. Yongxin Zhang
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Correspondence to Yongxin Zhang.

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A project supported by Scientific Research Fund of Sichuan Provincial Education Department(08ZA132).

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Zhang, Y. Homoclinic Solutions for a Forced Liénard Type System. Results. Math. 57, 69–78 (2010). https://doi.org/10.1007/s00025-009-0005-9

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  • DOI: https://doi.org/10.1007/s00025-009-0005-9

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