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 condition for a center Weak center Liapunov’s constants Liapunov's constants Isochronous center Darboux’s first integral Darboux's first integral Curves Center-focus problem Analytic planar differential system
Abstract: © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature. We consider analytic or polynomial vector fields of the form X=(-y+X)∂∂x+(x+Y)∂∂y, where X= X(x, y)) and Y= Y(x, y)) start at least with terms of second order. It is well-known that X has a center at the origin if and only if X has a Liapunov–Poincaré local analytic first integral of the form H=12(x2+y2)+∑j=3∞Hj, where H j = H j (x, y) is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincaré consists in distinguishing when the origin of X is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field X for which H is a first integral. Moreover, given an analytic function V=1+∑j=1∞Vj in a neighborhood of the origin, where V j is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field X for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form H=12(x2+y2)(1+∑j=1∞Υj), in a neighborhood of the origin, where Υ j is a homogenous polynomial of degree j for j≥ 1. These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin We extended to analytic or polynomial differential systems the weak conditions of a center given by Alwash and Lloyd for linear centers with homogeneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers.
Thematic Areas: Mathematics (miscellaneous) Mathematics (all) Mathematics Matemática / probabilidade e estatística General mathematics Engenharias iv
licence for use: https://creativecommons.org/licenses/by/3.0/es/
ISSN: 0009725X
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/submittedVersion
Link to the original source: https://link.springer.com/article/10.1007/s12215-018-0342-1
Papper original source: Rendiconti Del Circolo Matematico Di Palermo. 68 (1): 29-64
APA: Llibre J; Ramírez R; Ramírez V (2019). An inverse approach to the center problem. Rendiconti Del Circolo Matematico Di Palermo, 68(1), 29-64. DOI: 10.1007/s12215-018-0342-1
Licence document URL: https://repositori.urv.cat/ca/proteccio-de-dades/
Article's DOI: 10.1007/s12215-018-0342-1
Entity: Universitat Rovira i Virgili
Journal publication year: 2019
Publication Type: Journal Publications
Repositori URV