On the configurations of centers of planar Hamiltonian Kolmogorov cubic polynomial differential systems

Jaume Llibre and Dongmei Xiao

Vol. 306 (2020), No. 2, 611–644

DOI: 10.2140/pjm.2020.306.611

Abstract

We study the kind of centers that Hamiltonian Kolmogorov cubic polynomial differential systems can exhibit. Moreover, we analyze the possible configurations of these centers with respect to the invariant coordinate axes, and obtain that the real algebraic curve $xy\left(a+bx+cy+d{x}^2+exy+f{y}^2\right)=h$ has at most four families of level ovals in ${ℝ}^2$ for all real parameters $a,b,c,d,e,f$ and $h$.

Keywords

Hamiltonian system, Kolmogorov systems, cubic polynomial differential systems, centers, configuration of centers

Mathematical Subject Classification 2010

Primary: 37K10, 37C27

Secondary: 37K05

Milestones

Received: 13 December 2017

Revised: 2 April 2019

Accepted: 4 February 2020

Published: 13 July 2020