Limit Cycle Bifurcations in a Liénard System with a Hyperelliptic Gourd-Shaped Hamiltonian

• Qingying Xu ^[1] ; Liqin Zhao ^[1]
  1. [1] Beijing Normal University

     Beijing Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates the number of limit cycles in a hyperelliptic gourd-shaped Hamiltonian system under polynomial perturbations. Specifically, we consider the perturbed system ˙ x = y, ˙y = x − 5 4 x3 + 1 4 x5 + ε f (x, δ)y, where 0 < |ε| 1 and f (x, δ) = 6 i=0 ai xi is a polynomial of degree six. By studying limit cycles bifurcating near the elementary centers, the double homoclinic loop and the heteroclinic loop, we obtain lower bounds for the number of limit cycles and their corresponding configurations. Furthermore, under the assumption that some parameters ai vanish, we derive sharp upper bounds for the number of limit cycles by constructing ECT-systems and using geometric methods.

• Referencias bibliográficas
  • 1. Arnold, V.I.: Arnold’s Problems. Springer-Verlag, Berlin (2005)
  • 2. Asheghi, R., Zangeneh, R.: Bifurcations of limit cycles for a quintic Hamiltonian system with a double cuspidal loop. Comput.Math. Appl....
  • 3. Atabaigi, A., Zangeneh, R.: Bifurcation of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems of degree...
  • 4. Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four: (I) saddle loop and two saddle cycle. J. Differ. Equ....
  • 5. Gasull, A., Geyer, A., Mañosa, V.: Persistence of periodic traveling waves and Abelian integrals. J. Differ. Equ. 293, 48–69 (2021). https://doi.org/10.1016/j.jde.2021.05.033
  • 6. Grau,M., Mañosas, F., Villadelprat, J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011). https://doi.org/10.1090/S0002-9947-2010-05007-X
  • 7. Han, M., Yang, J., Tarta, A.A., Gao, Y.: Limit cycles near homoclinic and heteroclinic loops. J. Dyn. Differ. Equ. 20, 923–944 (2008)....
  • 8. Han, M., Yang, J., Yu, P.: Hopf bifurcations for near-Hamiltonian systems. Int. J. Bifurc. Chaos 19, 4117–4130 (2009). https://doi.org/10.1142/S0218127409025250
  • 9. Han, M.: Asymptotic expansions of melnikov functions and limit cycle bifurcations. Int. J. Bifurc. Chaos 22, 1250296 (2012). https://doi.org/10.1142/S0218127412502963
  • 10. Hu, Y., Niu,W., Huang, B.: Bounding the number of limit cycles for parametric Liénard systems using symbolic computation methods. Commun....
  • 11. Li, L., Yang, J.: On the number of limit cycles for a quintic Liénard system under polynomial perturbations. J. Appl. Anal. Comput. 9,...
  • 12. Mañosas, F., Villadelprat, J.: Bounding the number of zeros of certain Abelian integrals. J. Differ. Equ. 251, 1656–1669 (2011). https://doi.org/10.1016/j.jde.2011.05.026
  • 13. Pontryagin, L.: On dynamical systems close to Hamiltonian ones. Zh. Éksp. Teor. Fiz. 4, 234–238 (1934)
  • 14. Shao, Y.: Limit cycles bifurcations of a Liénard system with a hyperelliptic Hamiltonian of degree five. Electron. J. Qual. Theory Differ....
  • 15. Sun, X., Su, J., Han, M.: On the number of zeros of Abelian integral for some Liénard system of type (4,3). Chaos Solitons Fract. 51,...
  • 16. Sun, X., Xi, H., Zangeneh, R., Kazemi, R.: Bifurcation of limit cycles in small perturbation of a class of Liénard systems. Int. J. Bifurc....
  • 17. Sun, X., Yu, P.: Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree 4. J. Differ....
  • 18. Wang, J., Xiao, D.: On the number of limit cycles in small perturbations of a class of hyper-elliptic Hamiltonian systems with one nilpotent...
  • 19. Xu, W., Li, C.: Limit cycles of some polynomial Liénard systems. J. Math. Anal. Appl. 389, 367–378 (2012). https://doi.org/10.1016/j.jmaa.2011.11.070
  • 20. Yang, J., Han,M., Romanovski, V.G.: Limit cycle bifurcations of some Liénard systems. J. Math. Anal. Appl. 366, 242–255 (2010). https://doi.org/10.1016/j.jmaa.2009.12.035
  • 21. Yang, J., Liang, F.: Limit cycle bifurcations of a kind of Liénard system with a hyperbolic saddle and a nilpotent cusp. J. Appl. Anal....
  • 22. Zhao, L., Qi, M., Liu, C.: The cyclicity of period annuli of a class of quintic Hamiltonian systems. J. Math. Anal. Appl. 403, 391–407...
  • 23. Zhao, L.: The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle. Nonlinear...