A note on the Lyapunov and period constants
Cimà, Anna (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Mañosas Capellades, Francesc (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2020
Abstract:	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.
Grants:	Ministerio de Ciencia e Innovación MTM2016-77278-P Ministerio de Ciencia e Innovación MTM2017-86795-C3-1-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Limit cycle ; Critical period ; Bifurcation ; Lyapunov and period constants ; Cyclicity ; Criticality
Published in:	Qualitative theory of dynamical systems, Vol. 19, Issue 1 (April 2020) , art. 44, ISSN 1662-3592

DOI: 10.1007/s12346-020-00375-4

Postprint 9 p, 634.4 KB

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