Bifurcation of Limit Cycles from Quasi-Homogeneous Centers with Non-Smooth Perturbations

• Shengyu Chen ^[1] ; Shiyou Sui ^[1]
  1. [1] Tianjin University of Commerce

     Tianjin University of Commerce

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper estimates the number of limit cycles that bifurcate from periodic orbits of quasi-homogeneous centers under perturbations within the class of piecewise smooth polynomial systems of degree n. Using the first-order averaging method, we establish upper bounds for the number of limit cycles emerging from the period annuluses.

    As an application, the limit cycle bifurcation is studied for a quasi-homogeneous Hamiltonian system under piecewise smooth polynomial perturbations.

• Referencias bibliográficas
  • 1. Álvarez, M.J., Gasull, A., Prohens, R.: Global behaviour of the period function of the sum of two quasi-homogeneous vector fields. J. Math....
  • 2. Berezin, I.S., Zhidkov, N.P.: Computing Methods, vol. II. Pergamon Press, Oxford (1964)
  • 3. Cen, X., Liu, C., Yang, L., Zhang, M.: Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential...
  • 4. Françoise, J., He, H., Xiao, D.: The number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems...
  • 5. Françoise, J., Ji, H., Xiao, D., Yu, J.: Global dynamics of a piecewise smooth system for brain lactate metabolism. Qual. Theory Dyn. Syst....
  • 6. García, B., Llibre, J., Pérez del Río, J.: Planar quasi-homogeneous polynomial differential systems and their integrability. J. Differ....
  • 7. García, I.: On the integrability of quasihomogeneous and related planar vector fields. Int. J. Bifur. Chaos Appl. Sci. Engrg. 13, 995–1002...
  • 8. Gavrilov, L., Giné, J., Grau, M.: On the cyclicity of weight-homogeneous centers. J. Differ. Equ. 246, 3126–3135 (2009)
  • 9. Giné, J., Grau, M., Llibre, J.: Limit cycles bifurcating from planar polynomial quasi-homogeneous centers. J. Differ. Equ. 259, 7135–7160...
  • 10. Giné, J., Llibre, J., Valls, C.: Centers of weight-homogeneous polynomial vector fields on the plane. Proc. Amer.Math. Soc. 145, 2539–2555...
  • 11. Hilbert, D.: Mathematische problem. In: Lecture, Second Internat. Congr. Math., Paris (1990). in: Nachr. Ges. Wiss. Gottingen Math. Phys....
  • 12. Li, C., Liu, C., Yang, J.: A cubic system with thirteen limit cycles. J. Differ. Equ. 246, 3609–3619 (2009)
  • 13. Li, S., Wu, K.: On the limit cycles bifurcating from piecewise quasi-homogeneous differential center. Int. J. Bifur. Chaos Appl. Sci....
  • 14. Li, T., Chen, H., Chen, X.: Crossing periodic orbits of nonsmooth Li nard systems and applications. Nonlinearity 34, 4332–4372 (2021)
  • 15. Li, W., Llibre, J., Yang, J., Zhang, Z.: Limit cycles bifurcating from the period annulus of quasihomogeneous centers. J. Dyn. Differ....
  • 16. Lian, H., Liu, C., Yang, J.: On the cyclicity of quasi-homogeneous polynomial systems. J. Math. Anal. Appl. 516, 133–152 (2022)
  • 17. Prohens, R., Torregrosa, J.: New lower bounds for the Hilbert numbers using reversible centers. Nonlinearity 32, 331–355 (2019)
  • 18. Shi, S.: A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sin. 23, 153–158 (1980)
  • 19. Tang, Y.: Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete Contin....
  • 20. Teixeira, M.: Perturbation theory for non-smooth systems. In:Encyclopedia ofComplexity and Systems Science. Springer, New York (2009)
  • 21. Wei, L., Zhang, X.: Averaging theory of arbitrary order for piecewise smooth differential systems and its application. J. Dyn. Differ....
  • 22. Zhang, L.,Yu, J.:On the center criterion of planar quasi-homogeneous polynomial differential systems. Bull. Sci. Math. 147, 7–25 (2018)
  • 23. Zhao, Y., Zhang, Z.: Abelian integrals and period functions for quasihomogeneous Hamiltonian vector fields. Appl. Math. Lett. 18, 563–569...