Limit Cycles in Two Kinds of Quadratic Reversible Systems with Non-smooth Perturbations

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

     Ningxia Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 2, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00493-7
• Enlaces
  • Texto completo
• Resumen

  • This paper deals with the problem of limit cycle bifurcations for two kinds of quadratic reversible differential systems, when they are perturbed inside all discontinuous polynomials of degree n. The switching lines are x = 1 and y = 0. Firstly, we derive the algebraic structure of the first order Melnikov function M(h) by computing its generating functions, which is more complicated than the Melnikov function corresponding to the perturbations with one switching line. Then, we obtain the detailed expression of M(h) by solving the Picard–Fuchs equations that the generating functions satisfy. Finally, we derive the upper bounds of the number of limit cycles by using the derivation-division algorithm for n ≥ 2 and the lower bounds of the number of limit cycles by linear independence for n = 2, counting the multiplicity.

• Referencias bibliográficas
  • 1. X. Cen, C. Liu, L. Yang, M. Zhang, Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems,...
  • 2. B. Coll, A. Gasull, R. Prohens, Bifurcation of limit cycles from two families of centers, Dyn. Contin. Discrete Impuls. Syst. 12 (2005)...
  • 3. G. Dong, C. Liu, Note on limit cycles for m-piecewise discontinuous polynomial Liénard differential equations, Z. Angew. Math. Phys. 68...
  • 4. S. Gautier, L. Gavrilov, I.D. Iliev, Perturbations of quadratic centers of genus one, Discrete Contin. Dyn. Syst. 25(2) (2009) 511–535
  • 5. X. Hong, S. Xie, L. Chen, Estimating the number of zeros for Abelian integrals of quadratic reversible centers with orbits formed by higher-order...
  • 6. X. Hong, S. Xie, R. Ma, On the Ableian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree,...
  • 7. N. Hu, Z. Du, Bifurcation of periodic orbits emanated from a vertex in discontinuous planar systems, Commun. Nonlinear Sci. Numer. Simulat....
  • 8. M. Han, L. Sheng, Bifurcation of limit cycles in piecewise smooth systems via Melnikov function, J. Appl. Anal. Comput. 5 (2015) 809–815
  • 9. J. Itikawa, J. Llibre, A.Mereu, R. Oliveira, Limit cycles in uniform isochronous centers of discontinuous differential systems with four...
  • 10. Karlin, S., Studden, W.: Tchebycheff systems: with applications in analysis and statistics. Interscience Publishers (1966)
  • 11. F. Liang, M. Han, V. Romanovski, Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop,...
  • 12. X. Liu, M. Han, Bifurcation of limit cycles by perturbing piecewise Hamiltonian systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg 20...
  • 13. J. Llibre, J. Itikawa, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl....
  • 14. J. Llibre, A. Mereu, Limit cycles for discontinuous quadratic differetnial systems, J. Math. Anal. Appl. 413 (2014) 763–775
  • 15. J. Llibre, A. Mereu, D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations 258 (2015)...
  • 16. J. Shen, Z. Du, Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones, Z. Angew. Math. Phys. 67 (2016)...
  • 17. H. Tian, M. Han, Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems, J. Differential Equations...
  • 18. Y. Wang, M. Han, D. Constantinescu, On the limit cycles of perturbed discontinuous planar systems with 4 switching lines, Chaos Solitons...
  • 19. L. Wei, X. Zhang, Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differetnial systems, Discrete and Continuous...
  • 20. L. Wei, X. Zhang, Averaging theory of arbitrary order for piecewise smooth differential systems and its application, J. Dyn. Diff. Equat....
  • 21. Y. Xiong, M. Han, Limit cycle bifurcations in a class of perturbed piecewise smooth systems, Appl. Math. Comput. 242 (2014) 47–64
  • 22. Y. Xiong, M. Han, G. Romanovski (2017) The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with...
  • 23. Y. Xiong, Limit cycle bifurcations by perturbing non-smooth Hamiltonian systems with 4 switching lines via multiple parameters, Nonlinear...
  • 24. J. Yang, L. Zhao (2016) Limit cycle bifurcations for piecewise smooth Hamiltonian systems with a generalized eye-figure loop, International...
  • 25. J. Yang, L. Zhao, Limit cycle bifurcations for piecewise smooth integrable differential systems, Discrete and Continuous Dynamical Systems...
  • 26. J. Yang, L. Zhao, Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations, J. Differential...
  • 27. Zoł¸ ˙ adek, H.: Quadratic systems with center and their perturbations. J. Differ. Equ. 109:223–273 (1994)
  • 28. C. Zou, J. Yang, Piecewise linear differential system with a center-saddle type singularity, J. Math. Anal. Appl. 459 (2018) 453–463