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(a + bx + cy + dx2 + exy + fy2) = h has at most four families of level ovals in 2 for all real parameters a,b,c,d,e,f and h.

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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