An inverse approach to the center -focus problem for polynomial differential system with homogenous nonlinearities

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:263

Article

Llibre, Jaume¹  Ramirez, Rafael²  Ramirez, Valentin³ 

[1] Univ Autonoma Barcelona, Dept Matemat, E-08193 Barcelona, Catalonia, Spain

[2] Univ Rovira & Virgili, Dept Engn Informat & Matemat, Avinguda Paisos Catalans 26, E-43007 Tarragona, Catalonia, Spain

[3] Univ Barcelona, Gran Via Cortes Catalanas 585, E-08007 Barcelona, Spain

关键词: Center-foci problem;    Isochronous center;    Uniform isochronous center;    Holomorphic isochronous center;    Darboux's first integral;    Weak condition for a center;   

DOI  :  10.1016/j.jde.2017.04.030

来源: Elsevier

【 摘 要 】

We consider polynomial vector fields X with a linear type and with homogenous nonlinearities. 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=1/2(x(2) +y(2))+ Sigma(infinity)(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. Poincare 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 in such a way that H is a first integral of X and consequently the origin of X will be a center. We study the particular case of centers which have a local analytic first integral of the form H = 1/2(x(2)+y(2)) Sigma(infinity)(j=3) gamma(j), mial of degree gamma(j), for j >= 1 These centers are called weak centers, they contain the class of center studied by Alwash and Lloyd, the uniform isochronous centers and the isochronous holomorphic centers, but they do in a neighborhood of the origin, where gamma(j) is a convenient homogenous nonlinearities. (C)2017 Elsevier Inc. All rights reserved.

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