Limit Cycle Bifurcations in Perturbations of a Reversible Quadratic System with a Non-rational First Integral

• Xiong, Yanqin ^[1] ; Cheng, Rong ^[1] ; Li, Na ^[2]
  1. [1] Nanjing University of Information Science and Technology

     Nanjing University of Information Science and Technology

     China

     
  2. [2] Shanghai University of Engineering Science
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 3, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00434-w
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates the Hopf cyclicity of a piecewise smooth quadratic polynomial system by Melnikov function method, whose unperturbed system is a concrete reversible quadratic system with a center at the origin and with a non-rational first integral. By comparing the obtained result for the piecewise case with the result for the smooth case, it shows that the piecewise system can have at least four more limit cycles around the origin than the smooth one.

• Referencias bibliográficas
  • 1. Chicone, C., Jacobs, M.: Bifurcation of limit cycles from quadratic isochronous. J. Differ. Equ. 91, 268–326 (1991)
  • 2. Chen, F., Li, C., Llibre, J., Zhang, Z.: A unified proof on the weak Hilbert 16th problem for n = 2. J. Differ. Equ. 221, 309–342 (2006)
  • 3. Cen, X., Liu, C., Yang, L., Zhang, M.: Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential...
  • 4. Cruz, L., Novaes, D., Torregrosa, J.: New lower bound for the Hilbert number in piecewise quadratic differential systems. J. Differ. Equ....
  • 5. Castillo, J., Llibre, J., Verduzco, F.: The pseudo-Hopf bifurcation for planar discontinuous piecewise linear differential systems. Nonlinear...
  • 6. Emilio, F., Enrique, P., Francisco, T.: Canonical discontinuous planar piecewise linear systems. SIAM J. Appl. Dyn. Syst. 11, 181–211 (2012)
  • 7. Gautier, S., Gavrilov, L., Iliev, I.: Perturbatins of quadratic centers of genus one. Discrete Contin. Dyn. Syst. 25, 511–35 (2009)
  • 8. Hilbert, D.: Mathematial problems (M. Newton, Transl.). Bull. Amer. Math. Soc. 8, 437–479 (1902)
  • 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., Romanovski, V.G.: On the number of limit cycles of polynomial Liénard systems. Nonlinear Anal. Real World Appl. 14, 1655–1668...
  • 11. Han, M., Sheng, L.: Bifurcation of limit cycles in piecewise smooth systems via Melnikov function. J. Appl. Anal. Comput. 5, 809–815 (2015)
  • 12. Iliev, I.: Perturbations of quadratic centers. Bull. Sci. Math. 122, 107–161 (1998)
  • 13. Li, J.: Limit cycles bifurcated from a reversible quadratic center. Qualit. Theory Dyn. Syst. 6, 205–215 (2005)
  • 14. Liu, C.: Limit cycles bifurcated from some reversible quadratic centers with a non-algebraic first integral. Nonlinearity 25, 1653–1660...
  • 15. Llibre, J., Ramírez, R., Ramírez, V., Sadovskaia, N.: The 16th Hilbert problem restricted to circular algebraic limit cycles. J. Differ....
  • 16. Llibre, J., Tian, Y.: Algebraic limit cycles bifurcating from algebraic ovals of quadratic centers. Int. J. Bifurcat. Chaos 28, 1850145...
  • 17. Llibre, J., Tang, Y.: Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete...
  • 18. Li, C., Llibre, J.: A uniform study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems. J. Dyn. Differ....
  • 19. Li, C., Li, W., Llibre, J., Zhang, Z.: Linear estimate for the number of zeros of Abelian integrals for quadratic isochronous centres....
  • 20. Loud, W.: Behavior of the period of solutions of certain plane autonomous systems near centers. Contrib. Differ. Equ. 3, 21–36 (1964)
  • 21. Peng, L., Gao, Y., Feng, Z.: Limit cycles bifurcating from piecewise quadratic systems separated by a straight line. Nonlinear Anal. 196,...
  • 22. Schlomiuk, D.: Algebraic particular integrals integrability and the problem of the center. Trans. Am. Math. Soc. 338, 737–754 (1993)
  • 23. Tian, Y., Han, M.: Hopf and homoclinic bifurcations for near-Hamiltonian systems. J. Differ. Equ. 262, 3214–3234 (2017)
  • 24. Xiong, Y., Han, M.: Limit cycle bifurcations in a class of perturbed piecewise smooth systems. Appl. Math. Comput. 242, 47–64 (2014)
  • 25. Xiong, Y., Han, M.: New lower bounds for the Hilbert number of polynomial systems of Liénard type. J. Differ. Equ. 257, 2565–2590 (2014)
  • 26. Xiong, Y., Han, M., Romanovski, V.G.: The maximal number of limit cycles in perturbations of piecewise linear Hamiltonian systems with...
  • 27. Xiong, Y.: Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters. J. Math. Anal. Appl....
  • 28. Xiong, Y., Han, M.: Limit cycle bifurcations in a class of near-Hamiltonian systems with multiple parameters. Chaos Solitons Fractals...
  • 29. Yang, J., Yu, P.: Nine limit cycles around a singular point by perturbing a cubic Hamiltonian system with a nilpotent center. Appl. Math....
  • 30. Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations. J. Differ....
  • 31. Zoł¸adek, H.: Quadratic systems with center and their perturbations. J. Differ. Equ. 109, 223–273 (1994)
  • 32. Zoł¸adek, H.: Melnikov functions in quadratic perturbations of generalized Lotka-Volterra systems. J. Dyn. Control Syst. 21, 573–603 (2015)