Autor según el artículo: Llibre J; Ramírez R; Ramírez V

Departamento: Enginyeria Informàtica i Matemàtiques

Autor/es de la URV: Ramírez Inostroza, Rafael Orlando

Palabras clave: 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

Resumen: © 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.

Áreas temáticas: Mathematics (miscellaneous) Mathematics (all) Mathematics Matemática / probabilidade e estatística General mathematics Engenharias iv

Acceso a la licencia de uso: https://creativecommons.org/licenses/by/3.0/es/

ISSN: 0009725X

Direcció de correo del autor: rafaelorlando.ramirez@urv.cat

Identificador del autor: 0000-0002-4958-0291

Fecha de alta del registro: 2023-04-01

Versión del articulo depositado: info:eu-repo/semantics/submittedVersion

Enlace a la fuente original: https://link.springer.com/article/10.1007/s12215-018-0342-1

URL Documento de licencia: http://repositori.urv.cat/ca/proteccio-de-dades/

Referencia al articulo segun fuente origial: Rendiconti Del Circolo Matematico Di Palermo. 68 (1): 29-64

Referencia de l'ítem segons les normes 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

DOI del artículo: 10.1007/s12215-018-0342-1

Entidad: Universitat Rovira i Virgili

Año de publicación de la revista: 2019

Tipo de publicación: Journal Publications