Advertisement

Hopf Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

  • Ji Hua WangEmail author
Article
  • 3 Downloads

Abstract

This paper is concerned with small quadratic perturbations to one parameter family of generic reversible quadratic vector fields with a simple center. The first objective is to show that this system exhibits two small amplitude limit cycles emerging from a Hopf bifurcation. The second one we prove that the system has no limit cycle around the weak focus of order two. The results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev (1998).

Keywords

Quadratic reversible system limit cycle weak focus Hopf bifurcation 

MR(2010) Subject Classification

34C07 34C08 34C23 37G15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank the referees for their valuable comments and corrections which improve the presentation of the paper.

References

  1. [1]
    Arnold, V. I.: Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields. Funct. Anal. Appl., 11, 1–10 (1977)MathSciNetCrossRefGoogle Scholar
  2. [2]
    Artés, J., Llibre, J., Schlomiuk, D.: The geometry of quadratic differential systems with a weak focus of second order. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16(11), 3127–3194 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bautin, N.: On the number of limit cycles appearing with variations of coefficients from an equilibrium state of the type of a focus or a center, Mat. Sb. (N.S.) 30 (1952) 181–196; English Transl., Amer. Math. Sot. Transl. (1954)MathSciNetGoogle Scholar
  4. [4]
    Buica, A., Giné, J., Grau, M.: Essential perturbations of polynomial vector fields with a period annulus. Commun. Pure Appl. Anal., 14, 1073–1095 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Caubergh, M., Dumortie, F.: Hopf—Takens bifurcations and centres. J. Differential Equations, 202, 1–31 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Chen, G., Li, C., Liu, C., et al.: The cyclicity of period annuli of some classes of reversible quadratic systems. Discrete Contin. Dyn. Syst., 16, 157–177 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Chen, L., Wang, M.: The relative position and number of limit cycles of the quadratic differential system. Acta. Math. Sinica, Chin. Ser., 22(6), 751–758 (1979)MathSciNetzbMATHGoogle Scholar
  8. [8]
    Chicone, C., Jacobs, M.: Bifurcation of limit cycles from quadratic isochrones. J. Differential Equations, 91, 268–326 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Gavrilov, L.: The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math., 143, 449–497 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Han, M.: The Hopf cyclicity of Lienard systems. Appl. Math. Lett., 14, 183–188 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Huang, J., Liu, C., Wang, J.: Abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts. J. Math. Anal. Appl., 449(1), 884–896 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Iliev, I. D.: Perturbations of quadratic centers. Bull. Sci. Math., 122, 107–161 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Ilyashenko, Yu.: Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc., 39(3), 301–354 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Li, C.: Non-existence of limit cycle around a weak focus of order three for any quadratic system. Chin. Ann. of Math., 7(B), 174–190 (1986)zbMATHGoogle Scholar
  15. [15]
    Li, C.: Two problems of planar quadratic systems. Sci. China Ser. A, 26(5), 471–481 (1983)MathSciNetzbMATHGoogle Scholar
  16. [16]
    Li, C.: Abelian integrals and limit cycles. Qual. Theory Dyn. Syst., 11, 111–128 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Liu, J.: On the transformations and applications of quadratic systems on the plane. J. Wuhan Inst. Iron and Steel Tech., 4, 10–15 (1979)Google Scholar
  18. [18]
    Mieussens, M.: Sur les cycles limites des systemes quadratiques. C. R. Acad. Sc. Paris Ser. A., 291, 337–340 (1980)MathSciNetzbMATHGoogle Scholar
  19. [19]
    Roussarie, R.: Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem, Progress in Mathematics, 164, Birkhäuser Verlag, Basel, 1998CrossRefzbMATHGoogle Scholar
  20. [20]
    Sang, B., Zhu, S.: Focal quantities algorithms and center conditions deduction. Acta Math. Sci. Ser. A Chin. Ed., 28A(1), 164–173 (2008)zbMATHGoogle Scholar
  21. [21]
    Shi, S.: A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. China Ser. A., 23(2), 153–158 (1980)MathSciNetzbMATHGoogle Scholar
  22. [22]
    Wang, J., Dai, Y.: Cyclicity of a family of generic reversible quadratic systems with one center. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29(3), 1950035 (16 pages) (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Zeng, X., Zhang, Z., Gao, S.: On the uniqueness of the limit cycles of the generalized Liénard equation. Bull. London Math. Soc., 26, 213–247 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Zhang, P.: Quadratic systems with a 3rd-order (or 2nd-order) weak focus. Ann. Differential Equations, 17, 287–294 (2001)MathSciNetzbMATHGoogle Scholar
  25. [25]
    Zhao, Y.: On the number of limit cycles in quadratic perturbations of quadratic codimension four centers. Nonlinearity, 24, 2505–2522 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Zolądek, H.: Abelian integrals in unfoldings of codimension 3 singular planar vector fields, in “Lecture Notes in Mathematics, Vol. 1480,” pp. 165–224, Springer-Verlag, New York/Berlin (1991)Google Scholar
  27. [27]
    Zolądek, H.: Quadratic systems with center and their perturbations. J. Differential Equations, 109, 223–273 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Zolądek, H.: Melnikov functions in quadratic perturbations of generalized Lotka—Volterra systems. J. Dyn. Control Syst., 21, 573–603 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-sen UniversityGuangzhouP. R. China

Personalised recommendations