The Exact Number of Limit Cycles in a Class of Cubic Isochronous Hamiltonian Systems

• 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 5, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates the limit cycle problem for a class of cubic isochronous Hamiltonian systems, when they are perturbed inside of polynomials of degree n. A upper bound of the number of limit cycles is obtained using the Abelian integral. Moreover, this bound is sharp. The method for obtaining the algebraic structure of the Abelian integral differs from those in other literature.

• Referencias bibliográficas
  • 1. Anacleto, M., Vidal, C.: Limit cycles bifurcating of discontinuous and continuous piecewise differential systems of isochronous cubic centers...
  • 2. Arnold, V.: Ten problems in: Theory of singularities and its applications, Adv. Soviet Math. 1, 1–8 (1990)
  • 3. Binyamini, G., Novikov, D., Yakovenko, S.: On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal Hilbert...
  • 4. Calogero, F.: Isochronous Systems, Oxford University Press, (2008)
  • 5. Calogero, F., Leyvraz, F.: Isochronous and partially isochronous Hamiltonian systems are not rare. J. Math. Phys. 47, 042901 (2006)
  • 6. Calogero, F., Leyvraz, F.: General technique to produce isochronous Hamiltonians. J. Phys. A: Math. Theor. 40, 12931–12944 (2007)
  • 7. Calogero, F., Leyvraz, F.: Examples of isochronous Hamiltonians: classical and quantal treatments. J. Phys. A: Math. Theor. 41, 175202...
  • 8. Cen, X., Liu, C., Yang, L., Zhang, M.: Limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential...
  • 9. Chen, H., Tang, Y., Xiao, D.: Global dynamics of hybrid van der Pol-Rayleigh oscillators. Physica D 428, 133021 (2021)
  • 10. Chen, H., Zhang, R., Zhang, X.: Dynamics of polynomial Rayleigh-Duffing system near infinity and its global phase portraits with a center....
  • 11. Chang, Y., Zhao, L., Wang, Q.: The Poincaré bifurcation by perturbing a class of cubic Hamiltonian systems. Nonlinear Anal. RWA 82, 104246...
  • 12. Chen, Y., Yu, J.: The study on cyclicity of a class of cubic systems. Discrete Contin. Dyn. Syst. Ser. B 27(11), 6233–6256 (2022)
  • 13. Cima, A., Mañosas, F., Villadelprat, J.: Isochronicity for several classes of Hamiltonian systems. J. Differ. Equations 157, 373–413 (1999)
  • 14. Esteban, M., Llibre, J., Valls, C.: The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated...
  • 15. Franc ¸oise, J., Gavrilov, L.: Perturbation theory of the quadratic Lotka-Volterra double center. Commun. Contemp. Math. 24(5), 2150064...
  • 16. Gavrilov, L.: The infinitesimal 16th Hilbert problem in the quadratic case. Invent. Math. 143, 449–497 (2001)
  • 17. Guo, L., Wu, Y.: The isochronous centers for Kukles homogeneous system of degree eight. Appl. Math. Letters 144, 108700 (2023)
  • 18. Grau, M., Villadelprat, J.: Bifurcation of critical periods from Pleshkans isochrones. J. London Math. Soc. 81, 142–160 (2010)
  • 19. Horozov, E., Iliev, I.: Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians. Nonlinearity 11, 1521–1537...
  • 20. Iliev, I.: The cyclicity of the period annulus of the quadratic Hamiltonian triangle. J. Differ. Equations 128, 309–326 (1996)
  • 21. Li, C.: Abelian integrals and limit cycles. Qual. Theory Dyn. Syst. 11, 111–128 (2012)
  • 22. Li, C., Li, W., Llibre, J., Zhang, Z.: Linear estimate for the number of zeros of Abelian integrals for quadratic isochronous centers....
  • 23. Li, C., Zhang, Z.: Remarks on 16th weak Hilbert problem for n = 2. Nonlinearity 15, 1975–1992 (2002)
  • 24. Li, F., Liu, Y., Liu, Y., Yu, P.: Complex isochronous centers and linearization transformations for cubic Z2-equivariant planar systems....
  • 25. Li, Y., Shi, S., Zhang, J.: Isochronous global center of linear plus homogeneous polynomial systems and cubic systems. J. Differ. Equations...
  • 26. Liang, H., Li, S., Zhang, X.: Limit cycles and global dynamics of planar piecewise linear refracting systems of focus-focus type, Nonlinear...
  • 27. Llibre, J., Valls, C.: Global phase portraits of the generalized van der Pol systems. Bull. Sci. Math. 182, 103213 (2023)
  • 28. Llibre, J., Valls, C.: The phase portrait of all polynomial Liénard isochronous centers. Chaos, Solitons Fractals 180, 114500 (2024)
  • 29. Liu, S., Chen, X., Yao, C., Zhang, Z.: Stability of multiple attractors in the unidirectionally coupled circular networks of limit cycle...
  • 30. Marín, D., Villadelprat, J.: The criticality of reversible quadratic centers at the outer boundary of its period annulus. J. Differ. Equations...
  • 31. Novikov, D., Malev, S.: Linear estimate for the number of zeros of Abelian integrals. Qual. Theory Dyn. Syst. 16, 689–696 (2017)
  • 32. Saha, S., Gangopadhyay, G.: The existence of a stable limit cycle in the Lienard-Levinson-Smith (LLS) equation beyond the LLS Theorem....
  • 33. Strogatz, S.: SYNC: How Order Emerges from Chaos in the Universe. Grand Central Publishing, New York, Nature and Daily Life (2004)
  • 34. Sugie, J., Ishibashi, K.: Limit cycles of a class of Li nard systems derived from state-dependent impulses. Nonlinear Anal. Hybrid Syst...
  • 35. Sun, X., Yu, P.: Exact bound on the number of zeros of Abelian integrals for two hyper-elliptic Hamiltonian systems of degree 4. J. Differ....
  • 36. Wang, H., Li, J., Li, Z., et al.: The quantitative analysis of homoclinic orbits from quadratic isochronous systems. Commun. Nonlinear...
  • 37. Xiong, Y., Han, M.: On the limit cycle bifurcation of a polynomial system from a global center. Anal. Appl. 12(3), 251–268 (2014)
  • 38. Xiong, Y., Han, M.: Limit cycle bifurcations by perturbing a class of planar quintic vector fields. J. Differ. Equations 269, 10964–10994...
  • 39. Yang, J.: On the number of limit cycles for a perturbed cubic reversible Hamiltonian system. Chaos 34, 103122 (2024)
  • 40. Yang, J., Sui, S., Zhao, L.: On the number of zeros of Abelian integral for a class of cubic Hamilton systems with the phase portrait...
  • 41. Yang, J., Zhao, L.: The cyclicity of period annuli for a class of cubic Hamiltonian systems with nilpotent singular points. J. Differ....
  • 42. Yang, J., Zhao, L.: Bounding the number of limit cycles of discontinuous differential systems by using Picard-Fuchs equations. J. Differ....
  • 43. Yuan, Z., Ke, A., Han, M.: On the number of limit cycles of a class of Liénard-Rayleigh oscillators. Physica D 438, 133366 (2022)
  • 44. Zhao, Y., Zhang, Z.: Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians. J. Differ. Equations...