Limit Cycle Bifurcations from a Quadratic Center with Two Switching Lines

• Yang, Jihua ^[1]
  1. [1] Ningxia Normal University

     Ningxia Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00374-5
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study limit cycle bifurcations for a differential system with two switching lines by using Picard–Fuchs equation. We obtain a detailed expression of the corresponding first order Melnikov function which can be used to get the upper bound of the number of limit cycles. It is worth noting that we greatly simplify the computation. Our results also show that the number of switching lines has essential impact on the number of limit cycles bifurcating from a period annulus.

• Referencias bibliográficas
  • 1. Buica, A., Llibre, J.: Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128, 7–22 (2004)
  • 2. Bujac, C., Llibre, J., Vulpe, N.: First integrals and phase portraits of planar polynomial differential cubic systems with the maximum...
  • 3. Cen, X., Li, S., Zhao, Y.: On the number of limit cycles for a class of discontinuous quadratic differetnial systems. J. Math. Anal. Appl....
  • 4. Chen, H., Li, D., Xie, J., Yue, Y.: Limit cycles in planar continuous piecewise linear systems. Commun. Nonlinear Sci. Numer. Simul. 47,...
  • 5. di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems, Theory and Applications. Springer, London...
  • 6. Dong, G., Liu, C.: Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations. Z. Angew. Math. Phys....
  • 7. Gao, Y., Peng, L., Liu, C.: Bifurcation of limit cycles from a class of piecewise smooth systems with two vertical straight lines of singularity....
  • 8. Gentes, M.: Center conditions and limit cycles for the perturbation of an elliptic sector. Bull. Sci. Math. 133, 597–643 (2009)
  • 9. Han, M.: On the maximum number of periodic solutions of piecewise smooth periodic equations by average method. J. Appl. Anal. Comput. 7,...
  • 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. Hu, N., Du, Z.: Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems. Commun. Nonlinear Sci. Numer. Simul....
  • 12. Han, M., Sun, H., Balanov, Z.: Upper estimates for the number of periodic solutions to multidimensional systems. J. Differ. Equ. 266,...
  • 13. Itikawa, J., Llibre, J., Mereu, A.C., Oliveira, R.: Limit cycles in uniform isochronous centers of discontinuous differential systems...
  • 14. Krivan, V.: On the Gause predator-prey model with a refuge: a fresh look at the history. J. Theor. Biol. 274, 67–73 (2011)
  • 15. Li, S., Liu, C.: A linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system. J. Math....
  • 16. Liang, F., Han, M., Romanovski, V.: Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic...
  • 17. Liu, X., Han, M.: Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 20, 1379–1390...
  • 18. Llibre, J., Mereu, A.: Limit cycles for discontinuous quadratic differetnial systems. J. Math. Anal. Appl. 413, 763–775 (2014)
  • 19. Llibre, J., Mereu, A., Novaes, D.: Averaging theory for discontinuous piecewise differential systems. J. Differ. Equ. 258, 4007–4032 (2015)
  • 20. Li, Y., Yuan, L., Du, Z.: Bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with finitely many zones. Int....
  • 21. Shen, J., Du, Z.: Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones. Z. Angew. Math. Phys. 67,...
  • 22. Sanders, J., Vehrulst, F.: Averaging Method in Nonlinear Dynamical Systems, Applied Mathematical Sciences, vol. 59. Springer, Berlin (1985)
  • 23. Teixeira,M.: Perturbation theory for non-smooth systems. In: Encyclopedia of Complexity and Systems Science. Springer, New York (2009)
  • 24. Wang, Y., Han, M., Constantinescu, D.: On the limit cycles of perturbed discontinuous planar systems with 4 switching lines. Chaos Solitons...
  • 25. Xiong, Y.: Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters. Nonlinear...
  • 26. Xiong, Y., Hu, J.: Limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points....
  • 27. Xiong, Y., Han, M.: On the limit cycle bifurcation of a polynomial system from a global center. Anal. Appl. 12, 251–268 (2014)
  • 28. Yang, J., Zhao, L.: Limit cycle bifurcations for piecewise smooth Hamiltonian systems with a generalized eye-figure loop. Int. J. Bifurc....
  • 29. Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations. J. Differ....
  • 30. Yang, J., Zhao, L.: Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete Contin. Dyn. Syst. Ser. B...
  • 31. Zou, C., Yang, J.: Piecewise linear differential system with a center-saddle type singularity. J. Math. Anal. Appl. 459, 453–463 (2018)