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A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point

  • Feng Li [1] ; Ting Chen [3] ; Yuanyuan Liu [1] ; Pei Yu [2]
    1. [1] Linyi University

      Linyi University

      China

    2. [2] Western University

      Western University

      Azerbaiyán

    3. [3] Guangdong University of Finance and Economics
    Mostrar afiliaciones +
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
  • Idioma: inglés
  • Enlaces
  • Resumen

    • In this paper, we study the center-focus problem for a generalized cubic Kukles system with a nilpotent singular point, which consists of a cubic system with an extra 4th-order term. A complete classification is given on the center conditions which are explicitly expressed in term of the system parameters. A total of 15 cases are obtained, among them 4 for the generalized cubic Kukles system and 12 for the cubic Kukles system, with one common for both. One of the center conditions is analytic. Moreover, it is shown that 8 small-amplitude limit cycles can bifurcate in the neighborhood of the singular point for the generalized cubic Kukles system, while only 7 small-amplitude limit cycles can exist around the singular point for the cubic Kukles system. The center-focus problem for the generalized cubic Kukles system with a nilpotent origin is thoroughly solved

  • Referencias bibliográficas
    • 1. Alvarez, M.J., Gasull, A.: Monodromy and stability for nilpotent critical points. Int. J. Bifurc. Chaos 15, 1253–1265 (2005)
    • 2. Alvarez, M.J., Gasull, A.: Generating limits cycles from a nilpotent critical point via normal forms. J. Math. Anal. Appl. 318, 271–287...
    • 3. Amelbkin, V.V., Lukasevnky, N.A., Sadovski, A.P.: Nonlinear Oscillations in Second Order Systems. BGY Lenin. B.I. Press, Minsk (1992)....
    • 4. Bondar, Y.L., Sadovskii, A.P.: Variety of the center and limit cycles of a cubic system, which is reduced to Liénard form. Bul. Acad. ¸Stiin¸te...
    • 5. Chavarriga, J., Giacomin, H., Giné, J.: Local analytic integrability for nilpotent centers. Ergodic Theory Dyn. Syst. 23, 417–428 (2003)
    • 6. Cherkas, L.A.: Center conditions for some equations of the form yy = P(x) + Q(x)y + R(x)y2. Differ. Uravn. 8(8), 1435–1439...
    • 7. Cherkas, L.A.: On the conditions for a center for certain equations of the form yy = P(x)+ Q(x)y + R(x)y2. Differ. Equ. 8,...
    • 8. Christopher, C.J., Lloyd, N.G., Pearson, J.M.: On Cherkas’s method for centre conditions. Nonlinear World 2(4), 459–469 (1995)
    • 9. Christopher, C.J., Lloyd, N.G.: On the paper of Jin and Wang concerning the conditions for a centre in certain cubic systems. Bull. London...
    • 10. Colin, C.: An algebraic approach to the classification of centres in polynomial Liénard systems. J. Math. Anal. Appl. 229, 319–329 (1999)
    • 11. Giné, J.: Conditions for the existence of a center for the Kukles homogeneous systems. Comput. Math. Appl. 43(10–11), 1261–1269 (2002)
    • 12. Giné, J.: Center conditions for polynomial Liénard systems. Qual. Theory Dyn. Syst. 16(1), 119–126 (2017)
    • 13. Giné, J.: Center conditions for generalized polynomial Kukles systems. Commun. Pure Appl. Anal. 16(2), 417–425 (2017)
    • 14. Giné, J., Llibre, J., Valls, C.: Centers for the Kukles homogeneous systems with odd degree. Bull. London Math. Soc. 47(2), 315–324 (2015)
    • 15. Giné, J., Llibre, J., Valls, C.: Centers for the Kukles homogeneous systems with even degree. J. Appl. Anal. Comp. 7(4), 1534–1548 (2017)
    • 16. Hill, J.M., Lloyd, N.G., Pearson, J.M.: Algorithmic derivation of isochronicity conditions. Nonlinear Anal. 67, 52–69 (2007)
    • 17. Hill, J.M., Lloyd, N.G., Pearson, J.M.: Centres and limit cycles for an extended Kukles system. Electron. J. Differ. Equ. 119, 1–23 (2007)
    • 18. Hill, J.M., Lloyd, N.G., Pearson, J.M.: Limit cycles of a predator-prey model with intratrophic predation. J. Math. Anal. Appl. 349, 544–555...
    • 19. Kushner, A.A., Sadovskii, A.P.: Center conditions for Liénard-type systems of degree four. Vestn. Beloruss. Gos. Univ. Ser. 1 Fiz. Mat....
    • 20. Li, F., Li, S.: Integrability and limit cycles in cubic Kukles systems with a nilpotent singular point. Nonlinear Dyn. 96(1), 553–563...
    • 21. Li, F., Li, H.W., Liu, Y.Y.: New double bifurcation of nilpotent focus. Int. J. Bifurc. Chaos 31(4), 2150053 (2021)
    • 22. Li, F., Liu, Y., Liu, Y., Yu, P.: Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic...
    • 23. Linh, L.V., Sadovskii, A.P.: The centre-focus problem for analytical systems of Liénard form in degenerate case. Bul. Acad. Stiinte Repub....
    • 24. Liu, Y., Li, F.: Double bifurcation of nilpotent foci. Int. J. Bifurc. Chaos 25(3), 1550036 (2015)
    • 25. Liu, Y., Li, J.: New study on the center problem and bifurcations of limit cycles for the Lyapunov system II. Int. J. Bifurc. Chaos 19(09),...
    • 26. Liu, T., Liu, Y., Li, F.: A kind of bifurcation of limit cycle from a nilpotent critical point. J. Appl. Anal. Comp. 8, 10–18 (2018)
    • 27. Liu, T., Wu, L., Li, F.: Analytic center of nilpotent critical points. Int. J. Bifurc. Chaos 22(08), 1250198 (2012)
    • 28. Lloyd, N.G., Pearson, J.M.: Conditions for a centre and the bifurcation of limit cycles. In: Francoise, J.P., Roussarie, R. (eds) Bifurcations...
    • 29. Lloyd, N.G., Pearson, J.M.: Computing centre conditions for certain cubic systems. J. Comp. Appl. Math. 40, 323–336 (1992)
    • 30. Pearson, J.M., Lloyd, N.G.: Kukles revisited: Advances in computing techniques. Comp. Math. Appl. 60(10), 2797–2805 (2010)
    • 31. Pitchford, J.W., Brindley, J.: Intratrophic predation in simple predator-prey models. Bull. Math. Biol. 60, 937–953 (1998)
    • 32. Rebollo-Perdomo, S., Vidal, C.: Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete Contin. Dyn....
    • 33. Romanovski, V.G., Shafer, D.S.: The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhauser, Boston (2009)
    • 34. Sadovskii, A.P.: Solution of the center and foci problem for a cubic system of nonlinear oscillations. Differ. Uravn. 33(2), 236–244 (1997)....
    • 35. Sadovskii, A.P., Shcheglova, T.V.: Solution of the center-foci problem for a cubic system with nine parameters. Differ. Uravn. 47(2),...
    • 36. Sadovskii, A.P., Shcheglova, T.V.: Center conditions for a polynomial differential system. Differ. Uravn. 49(2),151164 (2013) (in Russian);...
    • 37. Yu, P., Li, F.: Bifurcation of limit cycles in a cubic-order planar system around a nilpotent critical point. J. Math. Anal. Appl. 453(2),...
    • 38. Zhang, Z., Ding, T., Huang, W., Dong, Z.: Qualitative Theory of Differential Equations. Science Press, Beijing (1997)
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