Highest weak focus order for trigonometric Liénard equations

  • Armengol Gasull
  • Jaume GinéEmail author
  • Claudia Valls


Given a planar analytic differential equation with a critical point which is a weak focus of order k,  it is well known that at most k limit cycles can bifurcate from it. Moreover, in case of analytic Liénard differential equations this order can be computed as one half of the multiplicity of an associated planar analytic map. By using this approach, we can give an upper bound of the maximum order of the weak focus of pure trigonometric Liénard equations only in terms of the degrees of the involved trigonometric polynomials. Our result extends to this trigonometric Liénard case a similar result known for polynomial Liénard equations.


Trigonometric Liénard equation Weak focus Cyclicity 

Mathematics Subject Classification

Primary 34C07 Secondary 13H15 34C25 37C27 



The first author is partially supported by “Agencia Estatal de Investigación” and “Ministerio de Ciencia, Innovación y Universidades”, Grant number MTM2016-77278-P and AGAUR, Generalitat de Catalunya, grant 2017-SGR-1617. The second author is partially supported by a MINECO/FEDER grant number MTM2017-84383-P and an AGAUR grant number 2017SGR-1276. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.


  1. 1.
    Alwash, M.A.M.: On a condition for a centre of cubic nonautonomous equations. Proc. R. Soc. Edinb. Sect. A 113, 289–291 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maĭer, A.G.: Theory of bifurcations of dynamic systems on a plane. Translated from the Russian. Halsted Press (A division of John Wiley & Sons), New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, (1973). xiv+482 ppGoogle Scholar
  3. 3.
    Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.: Singularités des applications différentiables. Mir, Moscow (1982)Google Scholar
  4. 4.
    Belykh, V.N., Pedersen, N.F., Soerensen, O.H.: Shunted-Josephson-junction model. I. The autonomous case. Phys. Rev. B 16, 4853–4859 (1977)CrossRefGoogle Scholar
  5. 5.
    Briskin, M., Françoise, J.-P., Yomdin, Y.: The Bautin ideal of the Abel equation. Nonlinearity 11, 431–443 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chavarriga, J.: Integrable systems in the plane with center type linear part Appl. Math. (Warsaw) 22, 285–309 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cherkas, L.A.: Conditions for a Liénard equation to have a centre. Differ. Equ. 12, 201–206 (1976)zbMATHGoogle Scholar
  8. 8.
    Christopher, C.: An algebraic approach to the classification of centers in polynomial Liénard systems. J. Math. Anal. Appl. 229, 319–329 (1999)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Christopher, C.J., Lloyd, N.G.: Small-amplitude limit cycles in polynomial Liénard systems. NoDEA 3, 183–190 (1996)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Christopher, C.J., Lloyd, N.G., Pearson, J.M.: On a Cherkas’s method for centre conditions. Nonlinear World 2, 459–469 (1995)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Christopher, C., Lynch, S.: Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces. Nonlinearity 12, 1099–1112 (1999)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Cima, A., Gasull, A., Mañosas, F.: A simple solution of some composition conjectures for abel equations. J. Math. Anal. Appl. 398, 477–486 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cima, A., Gasull, A., Mañosas, F.: An explicit bound of the number of vanishing double moments forcing composition. J. Differ. Equ. 255, 339–350 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gasull, A., Geyer, A., Mañosas, F.: On the number of limit cycles for perturbed pendulum equations. J. Differ. Equ. 261, 2141–2167 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gasull, A., Giné, J., Valls, C.: Center problem for trigonometric Liénard systems. J. Differ. Equ. 263, 3928–3942 (2017)CrossRefGoogle Scholar
  16. 16.
    Gasull, A., Guillamon, A., Mañosa, V.: An explicit expression of the first Liapunov and period constants with applications. J. Math. Anal. Appl. 211, 190–212 (1997)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gasull, A., Torregrosa, J.: Small-amplitude limit cycles in Liénard systems via multiplicity. J. Differ. Equ. 159, 186–211 (1999)CrossRefGoogle Scholar
  18. 18.
    Gasull, A., Torregrosa, J.: A new approach to the computation of the Lyapunov constants, the geometry of differential equations and dynamical systems. Comput. Appl. Math. 20, 149–177 (2001)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Giné, J.: On some open problems in planar differential systems and Hilbert’s 16th problem. Chaos Solitons Fractals 31(5), 1118–1134 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Giné, J., Grau, M., Llibre, J.: Universal centres and composition conditions. Proc. Lond. Math. Soc. (3) 106(3), 481–507 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Giné, J., Santallusia, X.: Implementation of a new algorithm of computation of the Poincaré-Liapunov constants. J. Comput. Appl. Math. 166(2), 465–476 (2004)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hassard, B., Wan, Y.H.: Bifurcation formulae derived from center manifold theory. J. Math. Anal. Appl. 63, 297–312 (1978)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Inoue, K.: Perturbed motion of a simple pendulum. J. Phys. Soc. Jpn. 57, 1226–1237 (1988)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lichardová, H.: Limit cycles in the equation of whirling pendulum with autonomous perturbation. Appl. Math. 44, 271–288 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Morozov, A.D.: Quasi-conservative Systems. Cycles, Resonances and Chaos, World Scientific Series on Nonlinear Science. World Scientific Publishing, River Edge (1998)Google Scholar
  26. 26.
    Pearson, J.M., Lloyd, N.G., Christopher, C.J.: Algorithmic derivation of centre conditions. SIAM Rev. 38, 619–636 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Roussarie, R.: Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem. Birkhäuser Verlag, Basel (1998)CrossRefGoogle Scholar
  28. 28.
    Sanders, J.A., Cushman, R.: Limit cycles in the Josephson equation. SIAM J. Math. Anal. 17, 495–511 (1986)MathSciNetCrossRefGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Departament de MatemàticaUniversitat de LleidaLleidaSpain
  3. 3.Departamento de Matemática, Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal

Personalised recommendations