New families of global cubic centers

• Main Author: Llibre, Jaume
• Publication Date: 2024
• Other Authors: Serantola, Leonardo P. [UNESP]
• Format: Article
• Language: eng
• Source: Repositório Institucional da UNESP
• Download full: http://dx.doi.org/10.1007/s40863-024-00411-0; https://hdl.handle.net/11449/302129
• topic: Center; Cubic polynomial differential systems; Global center

An equilibrium point p of a differential system in the plane R2 is a center if there exists a neighbourhood U of p such that U\{p} is filled with periodic orbits. A difficult classical problem in the qualitative theory of differential systems in the plane R2 is the problem of distinguishing between a focus and a center. A global center is a center p such that R2\{p} is filled with periodic orbits. Another difficult problem in the qualitative theory of differential systems in R2 is to distinguish inside a family of centers the ones which are global. Lloyd, Pearson and Romanovsky characterized when the origin of coordinates is a center for the family of cubic polynomial differential systems x˙=y-Cx2+B+2Dxy+Cy2+Px3+Gx2y-H+3Pxy2+Ky3,y˙=-x+Dx2+E+2Cxy-Dy2-Kx3-H+3Px2y-Gxy2+Py3. Here we characterize when the origin of this family of differential system is a global center.