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Mathematics
Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems
R. Boukoucha University of Bejaia
Abstract:
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.
Received: 08.12.2020 Revised: 22.01.2021 Accepted: 05.02.2021
Citation:
R. Boukoucha, “Explicit form of first integral and limit cycles for a class of planar Kolmogorov systems”, Proceedings of the YSU, Physical and Mathematical Sciences, 55:1 (2021), 1–11
Linking options:
https://www.mathnet.ru/eng/uzeru826 https://www.mathnet.ru/eng/uzeru/v55/i1/p1
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