Limit Cycles Bifurcating from the Quasi-Homogeneous Polynomial Centers of Weight-Degree 2 Under Non-Smooth Perturbations

• Shiyou Sui ^[1] ; Yongkang Zhang ^[2] ; Baoyi Li ^[2]
  1. [1] Tianjin University of Commerce

     Tianjin University of Commerce

     China

     
  2. [2] Tianjin Normal University

     Tianjin Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We investigate the maximum number of limit cycles bifurcating from the period annulus of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic piecewise smooth polynomials. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. When the switching line is x = 0 or y = 0, we obtain the sharp bounds of the number of limit cycles for the perturbed systems by using the first order averaging method. Our results indicate that non-smooth systems can have more limit cycles than smooth ones, and the switching lines play an important role in the dynamics of non-smooth systems.

• Referencias bibliográficas
  • 1. Berezin, I.S., Zhidkov, N.P.: Computing Methods. II. Pergamon Press, Oxford (1964)
  • 2. Chen, F., Li, C., Llibre, J., Zhang, Z.: A unified proof on the weak Hilbert 16th problem for n=2. J. Differential Equations. 221,...
  • 3. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Springer-Verlag,...
  • 4. Hilbert, D.: Mathematische problem. Bul. Amer. Math. Soc. 8, 437–479 (1902)
  • 5. Planar quasi-homogeneous polynomial differential systems and their integrability: García, B., Llibre, J., Pérez del Río. J. J. Differential...
  • 6. García, I.: On the integrability of quasihomogeneous and related planar vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13,...
  • 7. Giné, J., Grau, M., Llibre, J.: Limit cycles bifurcating from planar polynomial quasi-homogeneous centers. J. Differential Equations. 259,...
  • 8. Giné, J., Llibre, J., Valls, C.: Centers of weight-homogeneous polynomial vector fields on the plane. Proc. Amer. Math. Soc. 145, 2539–2555...
  • 9. Llibre, J., Lopes, B.D., de Moraes, J.R.: Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers...
  • 10. Llibre, J., Mereu, A.: Limit cycles for discontinuous quadratic differential systems with two zones. J. Math. Anal. Appl. 413, 763–775...
  • 11. Llibre, J., Novaes, D.D.: Rodrigues, C: Averaging theory at any order for computing limit cycles of discontinuous piecewise differential...
  • 12. Llibre, J., Pessoa, C.: On the centers of the weight-homogeneous polynomial vector fields on the plane. J. Math. Anal. Appl. 359, 722–730...
  • 13. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13, 47–106...
  • 14. Li, S., Cen, X., Zhao, Y.: Bifurcation of limit cycles by perturbing piecewise smooth integrable nonHamiltonian systems. Nonlinear Anal....
  • 15. Li, W., Llibre, J., Yang, J., Zhang, Z.: Limit cycles bifurcating from the period annulus of quasihomogeneous centers. J. Dynam. Differential...
  • 16. Liu, X., Han, M.: Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg....
  • 17. Smale, S.: Mathematical problems for the next century. Math. Intell. 20, 7–15 (1998)
  • 18. Wei, L., Zhang, X.: Averaging theory of arbitrary order for piecewise smooth differential systems and its application. J. Dynam. Differential...
  • 19. Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations. J. Differential...