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

作者：Llibre Jaume^*; Ramirez Rafael; Ramirez Valentin

来源：Journal of Differential Equations, 2017, 263(6): 3327-3369.

DOI：10.1016/j.jde.2017.04.030

• Center-foci problem
• Isochronous center
• Uniform isochronous center
• Holomorphic isochronous center
• Darboux's first integral
• Weak condition for a center

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.

• 出版日期2017-9-15