Limit Cycles Appearing from the Perturbation of a Cubic Isochronous Center

• Jihua Yang ^[1] ; Qipeng Zhang ^[1]
  1. [1] Tianjin Normal University

     Tianjin Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 3, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • For the polynomial differential system x˙ = −y + i+ j=3 αi,j xi y j , y˙ = x + i+ j=3 βi,j xi y j , αi,j, βi,j ∈ R, Pleshkan (Differ. Equations, 1969) established that the origin is an isochronous center of this system if and only if it can be transformed into one of the canonical forms S∗1 , S∗ 2 , S∗3 or S∗ 4 . Except for case S∗1 , the bifurcations of limit cycles in these four types of isochronous differential systems remain unexplored. In this paper, we focus on the bifurcation of limit cycles for the system S∗ 2 under perturbations by an arbitrary polynomial vector field. By employing the Abelian integral, we derive an upper bound for the number of limit cycles that can emerge from such perturbations. The lower bounds are also provided for n = 1, 2, 3, 4, and numerical simulations are conducted.

• Referencias bibliográficas
  • 1. Calogero, F.: Isochronous Systems. Oxford University Press, (2008)
  • 2. Calogero, F., Leyvraz, F.: Isochronous and partially isochronous Hamiltonian systems are not rare. J. Math. Phys. 47, 042901 (2006)
  • 3. Calogero, F., Leyvraz, F.: General technique to produce isochronous Hamiltonians. J. Phys. A: Math. Theor. 40, 12931–12944 (2007)
  • 4. Calogero, F., Leyvraz, F.: Examples of isochronous Hamiltonians: classical and quantal treatments. J. Phys. A: Math. Theor. 41, 175202...
  • 5. Cen, X., Liu, C., Yang, L., Zhang, M.: Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential...
  • 6. Chavarriga, J., García, I.: Isochronous Centers of Cubic Reversible Systems. Springer, Berlin Heidelberg (1999)
  • 7. Chicone, C., Jacobs, M.: Bifurcation of limit cycles from quadratic isochrones. J. Differ. Equations 91, 268–326 (1991)
  • 8. Hilbert, D.: Mathematical problems. Bull. Amer. Math. Soc. 8, 437–479 (1902)
  • 9. Grau, M., Villadelprat, J.: Bifurcation of critical periods from Pleshkans isochrones. J. London Math. Soc. 81, 142–160 (2010)
  • 10. Gasull, A., Li, C., Torregrosa, J.: Limit cycles appearing from the perturbation of a system with a multiple line of critical points....
  • 11. Gasull, A., Li, W., Llibre, J., Zhang, Z.: Chebyshev property of complete elliptic integrals and its application to Abelian integrals....
  • 12. Li, C., Li, W., Llibre, J., Zhang, Z.: Linear estimate for the number of zeros of Abelian integrals for quadratic isochronous centers....
  • 13. Li, F., Liu, Y., Liu, Y., Yu, P.: Complex isochronous centers and linearization transformations for cubic Z2-equivariant planar systems....
  • 14. Li, F., Wu, Y., Yu, P.: Complete classification on center of cubic planar systems symmetric with respect to a straight line. Commun. Nonlinear...
  • 15. Liang, H., Llibre, J., Torregrosa, J.: Limit cycles coming from some uniform isochronous centers. Adv. Nonlinear Stud. 16(2), 197–220...
  • 16. Loud, W.: Behavior of the period of solutions of certain plane autonomous systems near centers. Contrib. Differ. Equations 3, 21–36 (1964)
  • 17. Pleshkan, I.: A new method of investigating the isochronicity of a system of two differential equations. Differ. Equations 5, 796–802...
  • 18. Yang, J.: Picard-Fuchs equation applied to quadratic isochronous systems with two switching lines. Int. J. Bifurcat. Chaos 30, 2050042...