On the class of two dimensional Kolmogorov systems

Rachid Boukoucha, Mouna Yahiaoui

Abstract


In this paper we charaterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form

\[

\left\{

\begin{array}{l}

x^{\prime }=x\left( P\left( x,y\right) +R\left( x,y\right) \ln \left\vert

\frac{A\left( x,y\right) }{B\left( x,y\right) }\right\vert \right) , \\

y^{\prime }=y\left( Q\left( x,y\right) +R\left( x,y\right) \ln \left\vert

\frac{A\left( x,y\right) }{B\left( x,y\right) }\right\vert \right) ,

\end{array}

\right.

\]

where $A\left(x,y\right)$, $B\left(x,y\right)$, $P\left( x,y\right)$, $Q\left(x,y\right)$, $R\left(x,y\right)$ are homogeneous polynomials of degree $a$, $a$, $n$, $n$, $m$ respectively. Concrete example exhibiting the applicability of our result is introduced.


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Published: 2019-02-08

How to Cite this Article:

Rachid Boukoucha, Mouna Yahiaoui, On the class of two dimensional Kolmogorov systems, Engineering Mathematics Letters, Vol 2019 (2019), Article ID 4

Copyright © 2019 Rachid Boukoucha, Mouna Yahiaoui. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Engineering Mathematics Letters

ISSN 2049-9337

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