Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems

R. Boukoucha

Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 55 (2021) no. 1, pp. 1-11

Résumé

In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form \begin{equation}\nonumber \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.

Détail

Keywords: Kolmogorov system, first integral, periodic orbits, limit cycle.