Bifurcation of critical periods from Pleshkan's isochrones
Grau, Maite (Universitat de Lleida. Departament de Matemàtica)
Villadelprat Yagüe, Jordi (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)

Date:	2010
Abstract:	Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities ℓ3. In this paper we prove that if we perturb any of these isochrones inside ℓ3, then at most two critical periods bifurcate from its period annulus. Moreover, we show that, for each k=0, 1, 2, there are perturbations giving rise to exactly k critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers ℓ2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in ℓ2. We prove that if we perturb three of them inside ℓ2, then at most one critical period bifurcates from its period annulus. In addition, for each k=0, 1, we show that there are perturbations giving rise to exactly k critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.
Grants:	Ministerio de Ciencia y Tecnología MTM-2008-00694 Agència de Gestió d'Ajuts Universitaris i de Recerca 2005/SGR-550 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410 Ministerio de Ciencia y Tecnología MTM-2008-03437
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Critical period ; Isochronous center ; Period function ; Bifurcation ; Unfolding
Published in:	Journal of the London Mathematical Society, Vol. 81, Issue 1 (February 2010) , p. 142-160, ISSN 1469-7750

DOI: 10.1112/jlms/jdp062

Postprint 19 p, 499.4 KB

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