#### Vol. 306, No. 2, 2020

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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
##### 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
##### Authors
 Jaume Llibre Departament de Matemàtiques Universitat Autònoma de Barcelona Barcelona Spain Dongmei Xiao School of Mathematical Sciences Shanghai Jiao Tong University Shanghai China