A Note on the Lyapunov and Period Constants

• Cima, A ^[1] ; Gasull, A ^[1] ; Mañosas, F ^[1]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00375-4
• Enlaces
  • Texto completo
• Resumen

  • It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations.

• Referencias bibliográficas
  • 1. Bautin, N.N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center...
  • 2. Blows, T.R., Lloyd, N.G.: The number of limit cycles of certain polynomial differential equations. Proc. R. Soc. Edinb. Sect. A 98, 215–239...
  • 3. Cima, A., Gasull, A., Mañosa, V., Mañosas, F.: Algebraic properties of the Liapunov and period constants. Rocky Mt. J. Math. 27, 471–501...
  • 4. Cima, A., Gasull, A., Mañosa, V.: Bifurcation of 2-periodic orbits from non-hyperbolic fixed points. J. Math. Anal. Appl. 457, 568–584...
  • 5. Chicone, C., Jacobs, M.: Bifurcation of critical periods for plane vector fields. Trans. Am. Math. Soc. 312, 433–486 (1989)
  • 6. Hervé, M.: Several Complex Variables. Oxford Univ. Press, Oxford (1963)
  • 7. Karlin, S., Studden, W.: Tchebycheff systems: with applications in analysis and statistics. In: Pure and Applied Mathematics, vol. XV,...
  • 8. Mardesic, P.: Chebyshev systems and the versal unfolding of the cusps of order n. Travaux en cours, Hermann, vol. 57 (1998)
  • 9. Roussarie, R.: Bifurcation of Planar Vector Fields and Hilbert’s Sixteenth Problem. Birkhäuser Verlag, Basel (1998)
  • 10. Zuppa, C.: Order of cyclicity of the singular point of Linéard’s polynomial vector fields. Bol. Soc. Bras. Mat. 12, 105–111 (1981)