An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Ramírez, Rafael Orlando (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)
Ramírez, Valentín (Universitat de Barcelona)

Date:	2017
Abstract:	We consider polynomial vector fields of the form \[ \X=(-y X_m) x (x Y_m) y, \] where X_m=X_m(x,y) and Y_m=Y_m(x,y) are homogenous polynomials of degree m. It is well--known that \X has a center at the origin if and only if \X has an analytic first integral of the form \[ H=12(x^2 y^2) _j=3^ H_j, \] where H_j=H_j(x,y) is a homogenous polynomial of degree j. The classical center-focus problem already studied by H. Poincar\'e consists in distinguishing when the origin of \X is either a center or a focus. In this paper we study the inverse center-focus problem. In particular for a given analytic function H defined in a neighborhood of the origin we want to determine the homogenous polynomials X_m and Y_m in such a way that H is a first integral of \X and consequently the origin of \X will be a center. Moreover, we study the case when \[H=12(x^2 y^2)(1 _j=1^ \Upsilon_j),\] where \Upsilon_j is a convenient homogenous polynomial of degree j for j 1. The solution of the inverse center problem for polynomial differential systems with homogenous nonlinearities, provides a new mechanism to study the center problem, which is equivalent to Liapunov's Theorem and Reeb's criterion.
Grants:	Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 316338 European Commission 318999 Ministerio de Economía y Competitividad MTM2013-40998-P Ministerio de Economía y Competitividad MTM2016-77278-P
Note:	Agraïments: The second author was partly supported by the Spanish Ministry of Education through projects TIN2011-27076-C03-01, TIN2014-57364-C2-1-R.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Analytic planar differential system ; Center-foci problem ; Darboux's first integral ; Holomorphic isochronous center ; Isochronous center ; Liapunov's constants ; Uniform isochronous center ; Weak condition for a center
Published in:	Journal of differential equations, Vol. 263 (2017) , p. 3327-3369, ISSN 1090-2732

DOI: 10.1016/j.jde.2017.04.030

Postprint 37 p, 871.8 KB

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