On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems
In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form $$ x^{\prime}=x\left(B_{1}(x,y)\, \ln \left| \frac{A_{3}(x,y)}{A_{4}(x,y)}\right| + B_{3}(x,y)\ln \left| \frac{A_{1}(x,y)}{A_{2}(x,y)}\right| \right), $$ $$ y^{\prime}=y\left(B_{2}...
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| Format: | Article |
| Language: | English |
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EJAAM
2021-11-01
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| Series: | E-Journal of Analysis and Applied Mathematics |
| Subjects: | |
| Online Access: | https://ejaam.org/articles/2021/10.2478-ejaam-2021-0001.pdf |
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| Summary: | In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form $$ x^{\prime}=x\left(B_{1}(x,y)\, \ln \left| \frac{A_{3}(x,y)}{A_{4}(x,y)}\right| + B_{3}(x,y)\ln \left| \frac{A_{1}(x,y)}{A_{2}(x,y)}\right| \right), $$ $$ y^{\prime}=y\left(B_{2}(x,y)\ln \left| \frac{A_{5}(x,y)}{A_{6}(x,y)}\right| + B_{3}(x,y)\ln \left| \frac{A_{1}(x,y)}{A_{2}(x,y)}\right| \right) $$ where $A_{1}\left( x,y\right) ,$ $A_{2}\left( x,y\right) ,$ $A_{3}\left( x,y\right) ,$ $A_{4}\left( x,y\right) ,$ $A_{5}\left( x,y\right) ,$ $A_{6}\left( x,y\right) ,$ $B_{1}\left( x,y\right) ,$ $B_{2}\left( x,y\right),$ $B_{3}\left( x,y\right) $ are homogeneous polynomials of degree $a,$ $a,$ $b,$ $b,$ $c,$ $c,$ $n,$ $n,$ $m$ respectively. Concrete example exhibiting the applicability of our result is introduced. |
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| ISSN: | 2544-9990 |