The Maximum Number of Small-Amplitude Limit Cycles in Liénard-Type Systems with Cubic Restoring Terms

• Hongwei Shi ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
• Idioma: inglés
• DOI: 10.1007/s12346-024-01075-z
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we investigate the small-amplitude limit cycles of two classes of Liénard systems of the form x˙ = y−F(x, μ), y˙ = −g(x), where the damping term F(x, μ)is either a polynomial or a rational function qn (x) pm(x), (qn(x) and pm(x) are polynomials in x with degrees n and m, respectively), μ represents coefficients and the cubic restoring term g(x) = x − x2 − 1 k x3 with a non-zero constant k. By utilizing Picard-Fuchs equation, we gain the upper bounds of the number of small-amplitude limit cycles of these two systems for any k = 0. Moreover, the smaller upper bounds are obtained for k < −4, k = −9 2 , and the upper bound is sharp if the damping term is polynomial.

• Referencias bibliográficas
  • 1. Blows, T.R., Lloyd, N.G.: The number of small-amplitude limit cycles of Liénard equations. Math. Proc. Cambridge Philos. Soc. 95, 359–366...
  • 2. Borodzik, M., Zoł¸ ˙ adek, H.: Small amplitude limit cycles for the polyomial Liénard system. J. Differ. Equ. 245, 2522–2533 (2008)
  • 3. Bostan, A., Dumas, P.: Wronskians and linear independence. Am. Math. Monthly 117, 722–727 (2010)
  • 4. Buzzi, C.A., Carvalho, Y.R., Gasull, A.: The local period function for Hamiltonian systems with applications. J. Differ. Equ. 280, 590–617...
  • 5. Chen, J., Tian, Y.: Maximum number of small limit cycles in some rational Liénard systems with cubic restoring terms. Int. J. Bifur. Chaos...
  • 6. Christopher, C.J., Lloyd, N.G.: Small-amplitude limit cycles in polynomial Liénard systems. NoDEA Nonlinear Differ. Equ. Appl. 3, 183–190...
  • 7. Christopher, C.J., Lynch, S.: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring...
  • 8. Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four. I. Saddle loop and two saddle cycle. J. Differ. Equ....
  • 9. Dumortier, F., Li, C.: Perturbations from an elliptic Hamiltonian of degree four. II. Cuspidal loop. J. Differ. Equ. 175, 209–243 (2001)
  • 10. Dumortier, F., Li, C.: Perturbation from an elliptic Hamiltonian of degree four. III. Global centre. J. Differ. Equ. 188, 473–511 (2003)
  • 11. Dumortier, F., Li, C.: Perturbation from an elliptic Hamiltonian of degree four. IV. Figure eight-loop. J. Differ. Equ. 188, 512–554 (2003)
  • 12. Gasull, A., Torregrosa, J.: Small-amplitude limit cycles in Liénard systems via multiplicity. J. Differ. Equ. 159, 186–211 (1999)
  • 13. Gavrilov, L., Iliev, I.D.: Quadratic perturbations of quadratic codimension-four centers. J. Math. Anal. Appl. 357, 69–76 (2009)
  • 14. Grau, M., Villadelprat, J.: Bifurcation of critical periods from Pleshkans isochrones. J. Lond. Math. Soc. 81, 142–160 (2010)
  • 15. Han, M.: Lyapunov constants and Hopf cyclicity of Liénard systems. Ann. Differ. Equ. 15, 113–126 (1999)
  • 16. Han, M.: Bifurcation theory of limit cycles. Science Press, Beijing (2013)
  • 17. Han, M., Yang, J.: The maximum number of zeros of functions with parameters and application to differential equations. J. Nonlinear Model....
  • 18. Ilyashenko, Y., Yakovenko, S.: Lectures on analytic differential equations. Graduate Studies in Mathematics. 86, American Mathematical...
  • 19. Jiang, J., Han, M.: Small-amplitude limit cycles of some Liénard-type systems. Nonlinear Anal. 71, 6373–6377 (2009)
  • 20. Karlin, S., Studden, W.J.: Tchebycheff systems: with applications in analysis and statistics. Pure and Applied Mathematics, Interscience...
  • 21. Li, N., Han, M., Romanovski, V.G.: Cyclicity of some Liénard systems. Commun. Pure Appl. Anal. 14, 2127–2150 (2015)
  • 22. Lloyd, N.G., Lynch, S.: Small amplitude limit cycles of certain Liénard systems. Proc. R. Soc. Lond. Ser. A. 418, 199–208 (1988)
  • 23. Mardeši´c, P.: Chebyshev systems and the versal unfolding of the cusp of order n. Travaux en cours, 57, Hermann, Paris (1998)
  • 24. Petrov, G.S.: Nonoscillation of elliptic integrals. Funktsional Anal. i Prilozhen. 24, 45–50 (1990)
  • 25. Shi, H., Liu, C.: Hopf Cyclicity of a Class of Liénard-Type Systems. J. Nonlinear Model. Anal. 5, 682–694 (2023)
  • 26. Sun, X., Yu, P.: Cyclicity of periodic annulus and Hopf cyclicity in perturbing a hyper-elliptic Hamiltonian system with a degenerate...
  • 27. Tian, Y., Han, M.: Hopf bifurcation for two types of Liénard systems. J. Differ. Equ. 251, 834–859 (2011)
  • 28. Tian, Y., Han, M., Xu, F.: Bifurcations of small limit cycles in Liénard systems with cubic restoring terms. J. Differ. Equ. 267, 1561–1580...
  • 29. Zhao, Y., Zhang, Z.: Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians. J. Differ. Equ. 155,...