Bifurcation of Limit Cycles from a Non-Hamiltonian Quadratic Integrable System with Homoclinic Loop

  • Yulin ZhaoEmail author
  • Huaiping Zhu
Part of the Fields Institute Communications book series (FIC, volume 64)


In this chapter, we study quadratic perturbations of a non-Hamiltonian quadratic integrable system with a homoclinic loop. We prove that the perturbed system has at most two limit cycles in the finite phase plane, and the bound is exact. The proof relies on an estimation of the number of zeros of related Abelian integrals.



Zhao was partially supported by NSF of China (No. 11171355) and the Program for New Century Excellent Talents of Universities of China.This research of Zhu was supported by Natural Sciences and Engineering Research Council (NSERC) of Canada.

Received 8/16/2010; Accepted 10/10/2011


  1. 1.
    M. Cauberg, F. Dumortier, Hopf-Takens bifurcations and centers. J. Differ. Equat. 202, 1–31 (2004)CrossRefGoogle Scholar
  2. 2.
    C.Chicone, M.Jacobs, Bifurcation of limit cycles from quadratic isochronous. J. Differ. Equat. 91, 268–326 (1991)CrossRefGoogle Scholar
  3. 3.
    G. Chen, C. Li, C. Liu, J. Llibre, The cyclicity of period annuli of some classes of reversible quadratic systems. Discrete Contin. Dyn. Syst. 16, 157–177 (2006)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Shui-Nee Chow, C. Li, Y. Yi, The cyclicity of period annuli of degenerate quadratic Hamiltonian systems with elliptic segment loops. Ergodic Theory Dynam. Syst. 22(2), 349–374 (2002)Google Scholar
  5. 5.
    F. Dumortier, Freddy, C. Li, Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differ. Equat.139, 146–193 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    F. Dumortier, R. Roussarie, Abelian integrals and limit cycles. J. Differ. Equat. 227, 116–165 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math. 143(3), 449–497 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    L. Gavrilov, E. Horozov, Limit cycles of perturbations of quadratic Hamiltonian vector fields. J. Math. Pures Appl. 72(9), 213–238 (1993)MathSciNetzbMATHGoogle Scholar
  9. 9.
    P. Hartman, Ordinary Differential Equations, 2nd edn. (Birkhäuser, Boston, 1982)zbMATHGoogle Scholar
  10. 10.
    E.Horozov, I.D.Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian systems. Proc. London Math. Soc. 69, 198–224 (1994)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Y. He, C. Li, On the number of limit cycles arising from perturbations of homoclinic loops of quadratic integrable systems. Differ. Equ. Dyn. Syst. 5(3–4), 303–316 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    I.D. Iliev, Higher order Melnikov functions for degenerate cubic Hamiltonians. Adv. Differ. Equat. 1(4), 689–708 (1996)MathSciNetzbMATHGoogle Scholar
  13. 13.
    I.D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian triangle. J. Differ. Equat. 128, 309–326 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    I.D. Iliev, Perturbations of quadratic centers. Bull. Sci. Math. 122, 107–161 (1998)MathSciNetCrossRefGoogle Scholar
  15. 15.
    I.D. Iliev, C. Li, J. Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops. Nonlinearity 18, 305–330 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    C. Li, J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems. J. Dynam. Differ. Equat. 16(2), 271–195 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    C. Li, Z. Zhang, A criterion for determining the monotonicity of ratio of two Abelinan integrals. J. Differ. Equat. 124, 407–424 (1996)CrossRefGoogle Scholar
  18. 18.
    W. Li, Y. Zhao, C. Li, Z. Zhang, Abelian integrals for quadratic centers having almost all their orbits formed by quartics. Nonlinearity 15(3), 863–885 (2002)MathSciNetCrossRefGoogle Scholar
  19. 19.
    H. Liang, Y. Zhao, On the period function of reversible quadratic centers with their orbits inside quartics. Nonlinear Anal. 71, 5655–5671 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    H. Liang, Y. Zhao, Quadratic perturbations of a class of a quadratic reversible systems with one center. Discrete Continuous Dyn. Syst. 27, 325–335 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    C. Liu, The cyclicity of period annuli of a class of quadratic reversible systems with two centers, J. Differ. Equat. 252, 5260–5273 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, Progress in Mathematics, vol. 164 (Birkhäuser, Basel, 1998)CrossRefGoogle Scholar
  23. 23.
    D.S. Shafer, A. Zegeling, Bifurcation of limit cycles from quadratic centers. J. Differ. Equat. 122, 48–70 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Y. Ye, Theory of limit cycles, Transl. Math. Monographs, vol. 66 (American Mathematical Society, Providence, 1984)Google Scholar
  25. 25.
    J. Yu, C. Li, Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one-homoclinic loop. J. Math. Anal. Appl. 269(1), 227–243 (2002)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Y. Zhao, On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity. J. Differ. Equat. 205, 329–364 (2005)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Y. Zhao, Z. Liang, G. Lu, The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-Morsean point. J. Differ. Equat. 162, 199–223 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Y. Zhao, S. Zhu, Perturbations of the non-generic Hamiltonian vector fields with hyperbolic segment. Bull. Sci. math. 125(2), 109–138 (2001)MathSciNetCrossRefGoogle Scholar
  29. 29.
    H. Zoladek, Quadratic systems with centers and their perturbations. J. Differ. Equat. 109, 223–273 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics, School of Mathematics and Computational ScienceSun Yat-sen UniversityGuangzhouP.R. China
  2. 2.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations