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Center Conditions for Nilpotent Singularities in the Plane Using Invariant Solutions

    1. [1] Universitat de Lleida

      Universitat de Lleida

      Lérida, España

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
  • Idioma: inglés
  • DOI: 10.1007/s12346-024-01109-6
  • Enlaces
  • Resumen

    • Recalling that at any regular point we always have a unique particular solution curve passing through it. In this work it is constructed such particular solution curve not passing through the nilpotent singularity but as close as we want to the singularity. By product the existence of such particular curve allows to use it to determine necessary conditions to have a center for nilpotent singularities in the plane. Several involve methods to solve the center problem are known all based in the existence of a change of variables and a scaling transformation of time bringing any differential system with a nilpotent center into a time-reversible system. Here we present a new algebraic method based on the existence of such particular solution curve not passing through the singular point and the involution associated to the nilpotent system with a center. The algebraic method needs the computation of this particular curve up to certain order, which can be done with the help of an algebraic manipulator. Finally a new algebraic method is derived computing the vanishing of a unique function which really gives a scalar method for computing the necessary conditions.

  • Referencias bibliográficas
    • 1. Algaba, A., García, C., Giné, J.: Geometric criterium in the center problem. Mediterr. J. Math. 13, 2593–2611 (2016)
    • 2. Algaba, A., García, C., Giné, J.: Nilpotent centres via inverse integrating factors. European J. Appl. Math. 27(5), 781–795 (2016)
    • 3. Algaba, A., García, C., Giné, J.: Center conditions to find certain degenerate centers with characteristic directions. Math. Comput. Simulation...
    • 4. Algaba, A., García, C., Giné, J., Llibre, J.: The center problem for Z2-symmetric nilpotent vector fields. J. Math. Anal. Appl. 466(1),...
    • 5. Algaba, A., García, C., Reyes,M.: The center problem for a family of systems of differential equations having a nilpotent singular point....
    • 6. Álvarez, M.J., Gasull, A.: Monodromy and stability for nilpotent critical points, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(4), 1253–1265...
    • 7. Álvarez, M.J., Gasull, A.: Generating limit cycles from a nilpotent critical points via normal forms. J. Math. Anal. Appl. 318, 271–287...
    • 8. Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Qualitative Theory of Second-order Dynamic Systems, p. 101. Wiley, New York-Toronto...
    • 9. Andreev, A.F.: Solution of the problem of the center and the focus in one case. (Russian) Akad. Nauk SSSR. Prikl. Mat. Meh 17, 333–338...
    • 10. Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Local analytic integrability for nilpotent centers. Ergodic Theory Dyn. Syst. 23(2),...
    • 11. Chavarriga, J., Giné, J., Sorolla, J.: Analytic integrability of a class of nilpotent cubic systems. Math. Comput. Simulation 59(6), 489–495...
    • 12. 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. Uravn....
    • 13. Christopher, C.J.: An algebraic approach to the classification of centres in polynomial Liénard systems. J. Math. Anal. Appl. 229, 319–329...
    • 14. Demina, M.V., Giné, J., Valls, C.: Puiseux integrability of differential equations. Qual. Theory Dyn. Syst. 21(2), 35–35 (2022)
    • 15. García, I.A., Giacomini, H., Giné, J.: Generalized nonlinear superposition principles for polynomial planar vector fields. J. Lie Theory...
    • 16. García, I.A., Giacomini, H., Giné, J., Llibre, J.: Analytic nilpotent centers as limits of nondegenerate centers revisited. J. Math. Anal....
    • 17. García, I.A.: Formal inverse integrating factors and the nilpotent center problem. Int. J. Bifur. Chaos Appl. Sci. Engrg. 26(1), 1650015–160023...
    • 18. García, I.A.: ilpotent centers. J. Diff. Equ. 260(6), 5356–5377 (2016)
    • 19. García, I.A., Giné, J.: Analytic nilpotent centers with analytic first integral. Nonlinear Anal. 72, 3732– 3738 (2010)
    • 20. García, I.A., Giné, J.: Center problem with characteristic directions and inverse integrating factors. Commun. Nonlinear Sci. Numer. Simul....
    • 21. García, I.A., Giné, J.: Characterization of centers by its complex separatrices, preprint, Universitat de Lleida, (2023)
    • 22. Gasull, A., Torregrosa, J.: Center problem for several differential equations via Cherkas’ method. J. Math. Anal. Appl. 228(2), 322–343...
    • 23. Giacomini, H., Giné, J., Llibre, J.: The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic...
    • 24. Giné, J.: Analytic integrability of nilpotent cubic systems with degenerate infinity, Internat. J. Bifur. Chaos Appl. Sci. Eng. 11(8),...
    • 25. Giné, J.: Analytic integrability and characterization of centers for nilpotent singular points. Z. Angew. Math. Phys. 55(5), 725–740 (2004)
    • 26. Giné, J.: On some open problems in planar differential systems and Hilbert’s 16th problem. Chaos Solitons Fractals 31(5), 1118–1134 (2007)
    • 27. Giné, J.: Center conditions for polynomial Liénard systems. Qual. Theory Dyn. Syst. 16(1), 119–126 (2017)
    • 28. Giné, J.: Stability condition for nilpotent singularities by its complex separatrices.Math Methods Appl. Sci. (2024) https://doi.org/10.1002/mma.10197
    • 29. Giné, J., Llibre, J.:Amethod for characterizing nilpotent centers. J. Math.Anal. Appl. 413(1), 537–545 (2014)
    • 30. Giné, J., Maza, S.: The reversibility and the center problem. Nonlinear Anal. 74(2), 695–704 (2011)
    • 31. Liu, Y., Li, J.: New study on the center problem and bifurcations of limit cycles for the Liapunov system (I), Internat. J. Bifur. Chaos...
    • 32. Liu, Y., Li, J.: New study on the center problem and bifurcations of limit cycles for the Liapunov system (II), Internat. J. Bifur. Chaos...
    • 33. Moussu, R.: Symétrie et forme normale des centres et foyers dégénérés. Ergodic Theory Dyn. Syst. 2, 241–251 (1982)
    • 34. Pessoa, C., Queiroz, L.: Monodromic nilpotent singular points with odd Andreev number and the center problem, Qual. Theory Dyn. Syst 21(4),...
    • 35. Strózyna, E., Zoladek, H.: The analytic and formal normal form for the nilpotent singularity. J. Diff. Equ. 179(2), 479–537 (2002)
    • 36. Takens, F.: Singularities of vector fields. Inst. Hautes Études Sci. Publ. Math. 43, 47–100 (1974)
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