Phase portraits of the Higgins-Selkov system
Consider the planar autonomous system \[ \frac{dx}{dt}= (1-x^\gamma), \quad \frac{dy}{dt} =\alpha y(xy^{\gamma - 1} -1), \tag{1} \] where \(\alpha\) is a real number and \(\gamma > 1\) an integer. In case \(\gamma = 2\), system (1) has been used by E.E. Selkov to model glycolysis, the corresponding phase portrait has been investigated completely by \textit{J. C. Artés} et al. [Chaos Solitons Fractals 114, 145--150 (2018; Zbl 1415.34058)] and \textit{R. Mohanasubha} et al. [Springer Proc. Math. Stat. 266, 75--103 (2018; Zbl 1419.34116)]. The authors of the paper under review determine the phase portraits of (1) for \( \gamma = 3,4,5,6 \) in the Poincaré disc. Especially they prove that if system (1) has a limit cycle then this cycle is unique.
scientific article; zbMATH DE number 7452629
scientific article