Bifurcation of Limit Cycles for a Kind of Piecewise Smooth Differential Systems with an Elementary Center of Focus-Focus Type

• Zheng Si ^[1] ; Liqin Zhao ^[1]
  1. [1] Beijing Normal University

     Beijing Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº Extra 1, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01138-1
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study the number of limit cycles H(n) bifurcating from the piecewise smooth system formed by the quadratic reversible system (r22) for y ≥ 0 and the cubic system x˙ = y 1 + ¯x2 + y2 , y˙ = −¯x 1 + ¯x2 + y2 for y < 0 under the perturbations of polynomials with degree n, where x¯ = x − 1. By using the first-order Melnikov function, it is proved that 2n +3 ≤ H(n) ≤ 2n +7 for n ≥ 3 and the results are sharp for n = 0, 1, 2.

• Referencias bibliográficas
  • 1. Arnold, V.I.: Arnold’s Problems. Springer-Verlag, Berlin (2005)
  • 2. Chen, X., Han, M.: A linear estimate of the number of limit cycles for a piecewise smooth nearHamiltonian system. Qual. Theory Dyn. Syst....
  • 3. Chen, X., Han, M.: Number of limit cycles from a class of perturbed piecewise polynomial systems. Int. J. Bifur. Chaos Appl. Sci. Eng....
  • 4. Coll, B., Gasull, A., Prohens, R.: Degenerate Hopf bifurcations in discontinuous planar systems. J. Math. Anal. Appl. 253, 671–690 (2001)
  • 5. Coll, B., Gasull, A., Prohens, R.: Bifurcation of limit cycles from two families of centers. Dyn. Contin. Discrete Impuls. Syst. Ser. A...
  • 6. de Carvalho, T., Llibre, J., Tonon, D.J.: Limit cycles of discontinuous piecewise polynomial vector fields. J. Math. Anal. Appl. 449, 572–579...
  • 7. Gautier, S., Gavrilov, L., Iliev, I.D.: Perturbations of quadratic centers of genus one. Discr. Contin. Dyn. Syst. 25, 511–535 (2009)
  • 8. Gong, S., Han, M.: An estimate of the number of limit cycles bifurcating from a planar integrable system. Bull. Sci. Math. 176, 103118...
  • 9. Grau, M., Mañosas, F., Villadelprat, J.: A Chebyshev criterion for Abelian integrals. Trans. Am. Math. Soc. 363, 109–129 (2011)
  • 10. Han, M., Sheng, L.: Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J. Appl. Anal. Comput. 5, 809–815 (2015)
  • 11. Han, M., Yang, J.: The maximum number of zeros of functions with parameters and application to differential equations. J. Nonlinear Model....
  • 12. Hong, L., Hong, X., Lu, J.: A linear estimation to the number of zeros for Abelian integrals in a kind of quadratic reversible centers...
  • 13. Kuznetsov, Y.A., Rinaldi, S., Gragnani, A.: One-parameter bifurcations in planar Filippov systems. Int. J. Bifur. Chaos Appl. Sci. Eng....
  • 14. Li, C.: Abelian integrals and limit cycles. Qual. Theory Dyn. Syst. 11, 111–128 (2012)
  • 15. Li, C., Zhang, Z.: Remarks on 16th weak Hilbert problem for n = 2. Nonlinearity 15, 1975–1992 (2002)
  • 16. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifur. Chaos Appl. Sci. Eng. 13, 47–106 (2003)
  • 17. Li, S., Chen, X., Zhao, Y.: Bifurcation of limit cycles by perturbing piecewise smooth integrable non-Hamiltonian systems. Nonlinear Anal....
  • 18. Li, W., Zhao, Y., Li, C., Zhang, Z.: Abelian integrals for quadratic centres having almost all their orbits formed by quartics. Nonlinearity...
  • 19. Liang, F., Han, M., Romanovski, V.G.: Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic...
  • 20. Liu, X., Han, M.: Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Int. J. Bifurc. Chaos 20, 1379–1390 (2010)
  • 21. Llibre, J., Mereu, A., Novaes, D.D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032...
  • 22. Novaes, D.D., Torregrosa, J.: On extended Chebyshev systems with positive accuracy. J. Math. Anal. Appl. 448, 171–186 (2017)
  • 23. Yang, J.: Bifurcation of limit cycles of the nongeneric quadratic reversible system with discontinuous perturbations. Sci China Math....
  • 24. Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations. J. Differ....
  • 25. Zhang, H., Xiong, Y.: Hopf bifurcations by perturbing a class of reversible quadratic systems. Chaos Solitons Fractals 170, 113309 (2023)
  • 26. Zhu, C., Tian, Y.: Limit cycles from Hopf bifurcation in nongeneric quadratic reversible systems with piecewise perturbations. Int. J....