The Cyclicity of a Class of Global Nilpotent Center Under Perturbations of Piecewise Smooth Polynomials with Four Zones∗

• Li Zou ^[1] ; Liqin Zhao ^[1]
  1. [1] Beijing Normal University

     Beijing Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 3, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study the bifurcation of limit cycles of near-Hamilton system with four zones separated by nonlinear switching curves. We derive the expression of the first order Melnikov function. As an application, we consider the cyclicity of the system x˙=y,y˙=−x2m−1, where (0, 0) is a global nilpotent center and 2≤m∈N+, under the perturbations of piecewise smooth polynomials with four zones separated by y=±kxm with k>0. By analyzing the first order Melnikov function, we obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results.

• Referencias bibliográficas
  • 1. Andrade, K., Cespedes, O., Cruz, D., Novaes, D.: Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear...
  • 2. Bastos, J., Buzzi, C.A., Llibre, J., Novaes, D.D.: Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold....
  • 3. Coll, B., Gasull, A., Prohens, R.: Bifurcation of limit cycles from two families of centers. Dyn. Contin. Discrete Impuls. Syst. Ser. A...
  • 4. di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems Theory and Applications. Springer, London (2008)
  • 5. Francoise, J.P., Ji, H., Xiao, D., Yu, J.: Global dynamics of a piecewise smooth system for brain Lactate metabolism. Qual. Theory Dyn....
  • 6. Grau, M., Mañosas, Villadelprat, F.J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011)
  • 7. Han, M.: Bifurcation Theory of Limit Cycles. Science Press, Beijing (2013)
  • 8. Han, M., Yang, J.: The maximum number of zeros of functions with parameters and application to differential equations. J. Nonlinear Model....
  • 9. Hilbert, D.: Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902)
  • 10. Li, J.: Hilberts 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 47–106 (2003)
  • 11. Li, S., Liu, C.: A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system. J. Math....
  • 12. Li, S., Cen, X., Zhao, Y.: Bifurcation of limit cycles by perturbing piecewise smooth integrable nonHamiltonian systems. Nonlinear Anal....
  • 13. Li, S., Llibre, J.: Canard limit cycles for piecewise linear Liénard systems with three zones. Int. J. Bifur. Chaos Appl. Sci. Engrg....
  • 14. Liang, F., Romanovski, V., Zhang, D.: Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular...
  • 15. Liu, X., Han, M.: Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20, 1379–1390...
  • 16. Llibre, J., Zhang, X.: Limit cycles for discontinuous planar piecewise linear differential systems separated by an algebraic curve. Int....
  • 17. Peng, L., Gao, Y., Feng, Z.: Limit cycles bifurcating from piecewise quadratic systems separated by a straight line. Nonlinear Anal. 196,...
  • 18. Ramirez, O., Alves, A.M.: Bifurcation of limit cycles by perturbing piecewise non-Hamiltonian systems with nonlinear switching manifold....
  • 19. Sabatini, M.: On the period function of x + f (x)x2 + g(x) = 0. J. Differ. Equ. 196, 151–168 (2004)
  • 20. Sui, S., Yang, J., Zhao, L.: On the number of limit cycles for generic Lotka-Volterra system and Bogdanov-Takens system under perturbations...
  • 21. Tang, S., Liang, J.: Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge. Nonlinear Anal. 76, 165–180...
  • 22. Teixeira,M.: Perturbation theory for non-smooth systems. In: Encyclopedia of Complexity and Systems Science. Springer, New York (2009)
  • 23. Tian, H., Han, M.: Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with switching curve. Discrete Contin. Dyn. Syst....
  • 24. Wang, Y., Han, M., Constantinescu, D.: On the limit cycles of perturbed discontinuous planar systems with 4 switching lines. Chaos Solitons...
  • 25. Wang, J., Zhao, L.: The cyclicity of period annulus of degenerate quadratic Hamiltonian systems with polycycles S(2) or S(3) under perturbations...
  • 26. Xiong, Y., Han, M.: Limit cycles appearing from a generalized heteroclinic loop with a cusp and a nilpotent saddle. J. Differ. Equ. 303,...
  • 27. Yang, J.: Limit cycles appearing from the perturbation of differential systems with multiple switching curves. Chaos Solitons Fractals...
  • 28. Zang, H., Han, M., Xiao, D.: On Melnikov functions of a homoclinic loop through a nilpotent saddle for planar near-Hamiltonian systems....