English

Global centers of a class of cubic polynomial differential systems

Dynamical Systems 2023-10-12 v1

Abstract

A difficult classical problem in the qualitative theory of differential systems in the plane R2\mathbb{R}^2 is the center-focus problem, i.e. to distinguish between a focus and a center. Another difficult problem is to distinguish inside a family of centers the ones which are global. A global center is a center pp such that R2{p}\mathbb{R}^2\setminus\{p\} is filled with periodic orbits. In this paper we classify the global centers of the family of real polynomial differential systems of degree 33 that in complex notation write iw˙=wA3w2A4w3A5w2wA6ww2, i\dot{w}=w-A_3\overline{w}^2-A_4w^3-A_5w^2\overline{w}-A_6w\overline{w}^2, where w=x+iyw=x+iy and AkCA_k\in\mathbb{C} for k=3,4,5,6k=3,4,5,6.

Keywords

Cite

@article{arxiv.2310.07454,
  title  = {Global centers of a class of cubic polynomial differential systems},
  author = {Jaume Llibre and Gabriel Rondón},
  journal= {arXiv preprint arXiv:2310.07454},
  year   = {2023}
}