Author, as appears in the article.: Llibre J; Ramírez R; Ramírez V
Department: Enginyeria Informàtica i Matemàtiques
URV's Author/s: Ramírez Inostroza, Rafael Orlando / Ramírez Pérez, Rebeca
Keywords: Weak center Reeb integrating factor Poincaré-liapunov theorem Linear type center Darboux first integral
Abstract: © 2018 Texas State University. Λ-Ω differential systems are the real planar polynomial differential equations of degree m of the form (Formula presented), 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. A planar vector field with linear type center can be written as a 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 article is to study the center problem for Λ-Ω systems of degree m with (Formula presented), 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 (Formula presented) 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 transformationsP (X, Y, t) → (−X, Y, −t). If (µ + (m − 2))(a21 + a22) = 0 and (Formula presented) = 0 then the origin is a weak center. We observe that the main difficulty in proving this result for m > 6 is related to the huge computations.
Thematic Areas: Mathematics, applied Mathematics Matemática / probabilidade e estatística Interdisciplinar Ensino Engenharias iv Ciências agrárias i Ciência da computação Astronomia / física Analysis
licence for use: https://creativecommons.org/licenses/by/3.0/es/
ISSN: 10726691
Author's mail: rafaelorlando.ramirez@urv.cat rebeca.ramirez@estudiants.urv.cat
Author identifier: 0000-0002-4958-0291
Record's date: 2024-06-28
Papper version: info:eu-repo/semantics/acceptedVersion
Link to the original source: https://ejde.math.txstate.edu/index.html
Licence document URL: http://repositori.urv.cat/ca/proteccio-de-dades/
Papper original source: Electronic Journal Of Differential Equations. 2018 (184): 1-23
APA: Llibre J; Ramírez R; Ramírez V (2018). Center problem for generalized Λ-Ω differential systems. Electronic Journal Of Differential Equations, 2018(184), 1-23
Entity: Universitat Rovira i Virgili
Journal publication year: 2018
Publication Type: Journal Publications
Repositori URV