Center problem for generalized Λ-Ω differential systems
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 Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2018
Abstract:	The Λ-Ω differential systems are the real planar polynomial differential equations of degree m of the form ˙ x = −y(1 + Λ) + xΩ, ˙ y = x(1 + Λ) + yΩ, where Λ = Λ(x,y) and Ω = Ω(x,y) are polynomials of degree at most m−1 such that Λ(0,0) = Ω(0,0) = 0. We study the center problem for these Λ-Ω systems. A planar vector field with linear type center can be written as an Λ-Ω system if and only if the Poincaré-Liapunov first integral is of the form F = 1 2 (x2 + y2)(1 + O(x,y)). The main objective of this paper is to study the center problem for Λ-Ω systems of degree m with Λ = µ(a2x − a1y), and Ω = a1x + a2y + m−1 ∑ j=2 Ωj, where µ, a1, a2 are constants and Ωj = Ωj(x,y) is a homogenous polynomial of degree j, for j = 2,. . . ,m−1. We prove the following results. Assuming that m = 2,3,4,5 and (µ + (m−2))(a2 1 + a2 2) ̸= 0 and m−2 ∑ j=2 Ωj ̸= 0 then the Λ-Ω system has a weak center at the origin if and only if these systems after a linear change of variables (x,y) −→ (X,Y ) are invariant under the transformations (X,Y,t) −→ (−X,Y,−t). If (µ + (m−2))(a2 1 + a2 2) = 0 and m−2 ∑ j=1 Ωj = 0 then the origin is a weak center. We observe that the main difficulty to prove this result for m > 6 is related with the huge computations.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 Ministerio de Educación y Ciencia TIN2014-57364-C2-1-R Ministerio de Educación y Ciencia TSI2007-65406-C03-01
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Darboux first integral ; Linear type centers ; Poincaré-Liapunov Theorem ; Reeb integrating factor ; Weak center
Published in:	Electronic journal of differential equations, Vol. 2018, Issue 184 (2018) , p. 1-23, ISSN 1072-6691

Adreça alternativa: https://ejde.math.txstate.edu/Volumes/2018/184/llibre.pdf

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