Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems
Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 55 (2021) no. 1, pp. 1-11
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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.
Keywords:
Kolmogorov system, first integral, periodic orbits, limit cycle.
@article{UZERU_2021_55_1_a0,
author = {R. Boukoucha},
title = {Explicit form of first integral and limit cycles for a class of planar {Kolmogorov} systems},
journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
pages = {1--11},
publisher = {mathdoc},
volume = {55},
number = {1},
year = {2021},
language = {en},
url = {http://geodesic.mathdoc.fr/item/UZERU_2021_55_1_a0/}
}
TY - JOUR AU - R. Boukoucha TI - Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems JO - Proceedings of the Yerevan State University. Physical and mathematical sciences PY - 2021 SP - 1 EP - 11 VL - 55 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/UZERU_2021_55_1_a0/ LA - en ID - UZERU_2021_55_1_a0 ER -
%0 Journal Article %A R. Boukoucha %T Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems %J Proceedings of the Yerevan State University. Physical and mathematical sciences %D 2021 %P 1-11 %V 55 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/UZERU_2021_55_1_a0/ %G en %F UZERU_2021_55_1_a0
R. Boukoucha. Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems. Proceedings of the Yerevan State University. Physical and mathematical sciences, Tome 55 (2021) no. 1, pp. 1-11. http://geodesic.mathdoc.fr/item/UZERU_2021_55_1_a0/