Analytically Integrable Centers of Perturbations of Cubic Homogeneous Systems

  1. Algaba, Antonio 1
  2. García, Cristóbal 1
  3. Reyes, Manuel 1
  1. 1 Universidad de Huelva
    info

    Universidad de Huelva

    Huelva, España

    ROR https://ror.org/03a1kt624

Journal:
Qualitative theory of dynamical systems

ISSN: 1575-5460

Year of publication: 2021

Volume: 20

Issue: 2

Type: Article

DOI: 10.1007/S12346-021-00479-5 DIALNET GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Qualitative theory of dynamical systems

Abstract

We consider the analytically integrable perturbations of cubic homogeneous differential systems whose origin is an isolated singularity. We prove that are orbitally equivalent to the cubic vector field associated. We also characterize the analytically integrable centers. We apply the results to two families of degenerate vector fields.

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Bibliographic References

  • Algaba, A., Checa, I., García, C., Giné, J.: Analytic integrability inside a family of degenerate centers. Nonlinear Anal. Real World Appl. 31, 288–307 (2016)
  • Algaba, A., Gamero, E., García, C.: The integrability problem for a class of planar systems. Nonlinearity 22, 395–420 (2009)
  • Algaba, A., Díaz, M., García, C., Giné, J.: Analytic integrability around a nilpotent singularity: the non-generic case. Comm. Pure Appl. Anal. 19, 407–423 (2020)
  • Algaba, A., García, C., Giné, J.: Analytic integrability around a nilpotent singularity. J. Differ. Equ. 267, 443–467 (2019)
  • Algaba, A., García, C., Reyes, M.: Analytic invariant curves and analytic integrability of a planar vector field. J. Differ. Equ. 266, 1357–1376 (2019)
  • Algaba, A., García, C., Reyes, M.: Analytical integrability of perturbations of quadratic systems. Mediterr. J. Math. (2021). https://doi.org/10.1007/s00009-020-01647-8
  • Algaba, A., García, C., Reyes, M.: Analytical integrability problem for perturbations of cubic Kolmogorov systems. Chaos, Solitons Fractals 113, 1–10 (2018)
  • Algaba, A., García, C., Reyes, M.: Like-linearizations of vector fields. Bull. Sci. Math. 133, 806–816 (2009)
  • Algaba, A., García, C., Reyes, M.: Integrability of two dimensional quasi-homogeneous polynomial differential systems. Rocky Mt. J. Math. 41(1), 1–22 (2011)
  • Basov, V.V.: Two-dimensional homogeneous cubic systems: classification and normal forms. I St. Petersburg University. Mathematics 49(2), 99–110 (2016)
  • Chen, X., Giné, J., Romanovski, V.G., Shafer, D.S.: The 1:-q resonant center problem for certain cubic Lotka-Volterra systems. Appl. Math. Comput. 218, 11620–11633 (2012)
  • Christopher, C., Mardešic, P., Rousseau, C.: Normalizable, integrable, and linearizable saddle points for complex quadratic systems in C2. J. Dynam. Control Syst. 9(3), 311–363 (2003)
  • Christopher, C., Rousseau, C.: Normalizable, integrable and linearizable saddle points in the LotkaVolterra system. Qual. Theory Din. Syst. 5(1), 11–61 (2004)
  • Ferˇcec, B., Giné, J.: A blow-up method to prove formal integrability for some planar differential systems. J. Appl. Anal. Math. Comput 8(6), 1833–50 (2018)
  • Ferˇcec, B., Giné, J.: Blow-up method to compute necessary conditions of integrability for planar differential systems Appl. Math. Comput 358, 16–24 (2019)
  • Giné, J., Romanovski, V.G.: Integrability conditions for Lotka-Volterra planar complex quintic systems. Nonlinear Anal. Real World Appl. 11(3), 2100–2105 (2010)
  • Han, M., Jiang, K.: Normal forms of integrable systems at a resonant saddle. Ann. Differ. Equ. 14(2), 150–155 (1998)
  • Liu, C., Chen, G., Chen, G.: Integrability of Lotka-Volterra type systems of degree 4. J. Math. Anal. Appl. 388(2), 1107–1116 (2012)
  • Liu, C., Chen, G., Li, C.: Integrability and linearizability of the Lotka-Volterra systems. J. Differ. Equ. 198(2), 301–320 (2004)
  • Mattei, J.. F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. (4) 13(4), 469–523 (1980)
  • Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles, J Math Pures Appl 4 (1885) 167–244; Oeuvres de Henri Poincaré, vol. I, pp. 95–114. Gauthier-Villars, Paris (1951)
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