EXPLICIT FORM FOR THE FIRST INTEGRAL AND LIMIT CYCLES OF A CLASS OF PLANAR KOLMOGOROV SYSTEMS
Open Access
- 21 May 2021
- journal article
- Published by Yerevan State University in Proceedings of the YSU A: Physical and Mathematical Sciences
- Vol. 55 (1 (254)), 1-11
- https://doi.org/10.46991/pysu:a/2021.55.1.01
Abstract
In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form \begin{equation*} \left\{ \begin{array}{l} x^{\prime }=x\left( R\left( x,y\right) \exp \left( \dfrac{A\left( x,y\right) }{B\left( x,y\right) }\right) +P\left( x,y\right) \exp \left( \dfrac{C\left( x,y\right) }{D\left( x,y\right) }\right) \right) , \\ \\ y^{\prime }=y\left( R\left( x,y\right) \exp \left( \dfrac{A\left( x,y\right) }{B\left( x,y\right) }\right) +Q\left( x,y\right) \exp \left( \dfrac{V\left( x,y\right) }{W\left( x,y\right) }\right) \right) , \end{array} \right. \end{equation*} where $A\left( x,y\right)$, $B\left( x,y\right)$, $C\left( x,y\right)$, $D\left( x,y\right)$, $P\left( x,y\right)$, $Q\left( x,y\right)$, $R\left(x,y\right)$, $V\left( x,y\right)$, $W\left( x,y\right)$ are homogeneous polynomials of degree $a$, $a$, $b$, $b$, $n$, $n$, $m$, $c$, $c$, respectively. Concrete example exhibiting the applicability of our result is introduced.