
A class of quintic Kolmogorov systems with explicit nonalgebraic limit cycle
Ahmed Bendjeddou^{a}, Mohamed Grazem^{b} ^{a} Department of Mathematics, Faculty of sciences,
University of setif 1, 19000,
Algeria
^{b} Department of Mathematics, Faculty of sciences,
University of Boumerdes, 35000, Algeria
Abstract:
Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Subsequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form
\begin{equation*} \begin{array}{c} \overset{.}{x} =x P_{4}( x,y),\overset{.}{y} =y Q_{4}( x,y), \end{array} \end{equation*} where $P_{4}$ and $Q_{4}$ are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant $\{ (x,y)\in\mathbb{R}^{2}; x>0, y>0\}$. According to the hypothesises, the existence of algebraic or nonalgebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result.
Keywords:
Kolmogorov systems, first integral, periodic orbits, algebraic and nonalgebraic limit cycle.
DOI:
https://doi.org/10.17516/199713972019123285297
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UDC:
517.9 Received: 26.11.2018 Received in revised form: 29.01.2019 Accepted: 06.02.2019
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Ahmed Bendjeddou, Mohamed Grazem, “A class of quintic Kolmogorov systems with explicit nonalgebraic limit cycle”, J. Sib. Fed. Univ. Math. Phys., 12:3 (2019), 285–297
Citation in format AMSBIB
\Bibitem{BenGra19}
\by Ahmed~Bendjeddou, Mohamed~Grazem
\paper A class of quintic Kolmogorov systems with explicit nonalgebraic limit cycle
\jour J. Sib. Fed. Univ. Math. Phys.
\yr 2019
\vol 12
\issue 3
\pages 285297
\mathnet{http://mi.mathnet.ru/jsfu756}
\crossref{https://doi.org/10.17516/199713972019123285297}
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