Advertisement

On a conjecture on the integrability of Liénard systems

  • Jaume LlibreEmail author
  • Adrian C. Murza
  • Claudia Valls
Article
  • 7 Downloads

Abstract

We consider the Liénard differential systems in \(\mathbb {C}^2\) where F(x) is an analytic function satisfying \(F(0)=0\) and \(F'(0)\ne 0\). Then these systems have a strong saddle at the origin of coordinates. It has been conjecture that if such systems have an analytic first integral defined in a neighborhood of the origin, then the function F(x) is linear, i.e. \(F(x)=a x\). Here we prove this conjecture, and show that when F(x) is linear and system (1) has an analytic first integral, this is a polynomial.

Keywords

Liénard system First integral Strong saddle 

Mathematics Subject Classification

Primary: 34C07 34C05 34C40 

Notes

Acknowledgements

The first author is partially supported by by the Ministerio de Econo–mía, Industria y Competitividad, Agencia Estatal de Investigación Grant MTM 2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca Grant 2017SGR1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. The second author acknowledges support from a grant of the Romanian National Authority for Scientific Research and Innovation project number PN-II-RU-TE-2014-4-0657. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.

References

  1. 1.
    Cai, S.L., Zhang, P.G.: A quadratic system with weak saddle II. Ann. Differ. Equ. 4(2), 131–142 (1988)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Fronville, A., Sadovski, A., Zoladek, H.: Solution of the \(1:-2\) resonant center problem in the quadratic case. Dedicated to the memory of Wieslaw Szlenk. Fund. Math. 157, 191–207 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Furta, S.D.: On non-integrability of general systems of differential equations. Z. Angew. Math. Phys. 47, 112–131 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gasull, A., Giné, J.: Integrability of Liénard systems with a weak saddle. Z. Angew. Math. Phys. 68(13), 7 (2017)zbMATHGoogle Scholar
  5. 5.
    Giné, J., Llibre, J.: On the integrability of Liénard systems with a strong saddle. Appl. Math. Lett. 70, 39–45 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Li, W., Llibre, J., Zhang, X.: Planar analytic vector fields with generalized rational first integrals. Bull. Sci. Math. 125, 341–361 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Liu, C., Chen, G., Li, C.: Integrability and linearizability of the Lotka–Volterra systems. J. Differ. Equ. 198, 301–320 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Liu, C., Li, Y.: The integrability of a class of cubic Lotka–Volterra systems. Nonlinear Anal. Real World Appl. 19, 67–74 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Llibre, J., Valls, C.: On the analytic integrability of the Liénard analytic differential systems. Disc. Cont. Dyn. Sys. 21, 557–573 (2016)zbMATHGoogle Scholar
  10. 10.
    Poincaré, H.: Sur l’intégration des équations différentielles du premier order et du premier degré I and II. Rendiconti del Circolo Matematico di Palermo 5, 161–191 (1891)CrossRefGoogle Scholar
  11. 11.
    Poincaré, H.: Sur l’intégration des équations différentielles du premier order et du premier degré I and II. Rendiconti del Circolo Matematico di Palermo 11, 193–239 (1897)CrossRefzbMATHGoogle Scholar
  12. 12.
    Wan, W., Chi, X.: The formula of generalized center-weak saddle. Northeast. Math. J. 18(1), 89–94 (2002)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zhang, Q., Liu, Y.: Integrability and generalized center problem of resonant singular point. Appl. Math. Lett. 40, 13–16 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Zhu, D.M.: Saddle values and integrability conditions of quadratic differential systems. A Chin. Summary Appears in Chin. Ann. Math. Ser. A 8(5), 645 (1987)MathSciNetGoogle Scholar
  15. 15.
    Zhu, D.M.: Saddle values and integrability conditions of quadratic differential systems. Chinese Ann. Math. Ser. B 8(4), 466–478 (1987)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Zoladek, H.: The problem of center for resonant singular points of polynomial vector fields. J. Differ. Equ. 137(1), 94–118 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2019

Authors and Affiliations

  • Jaume Llibre
    • 1
    Email author
  • Adrian C. Murza
    • 2
    • 3
  • Claudia Valls
    • 4
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  3. 3.Department of Mathematical SciencesUniversity of Texas at DallasRichardsonUSA
  4. 4.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisboaPortugal

Personalised recommendations