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 $$ x^{\prime}=x\left(B_{1}(x,y)\, \ln \left| \frac{A_{3}(x,y)}{A_{4}(x,y)}\right| + B_{3}(x,y)\ln \left| \frac{A_{1}(x,y)}{A_{2}(x,y)}\right| \right), $$ $$ y^{\prime}=y\left(B_{2}...
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EJAAM
2021-11-01
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| Series: | E-Journal of Analysis and Applied Mathematics |
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| Online Access: | https://ejaam.org/articles/2021/10.2478-ejaam-2021-0001.pdf |
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| author | Rachid Boukoucha |
| author_facet | Rachid Boukoucha |
| author_sort | Rachid Boukoucha |
| collection | DOAJ |
| description | In this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form $$ 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) $$ where $A_{1}\left( x,y\right) ,$ $A_{2}\left( x,y\right) ,$ $A_{3}\left( x,y\right) ,$ $A_{4}\left( x,y\right) ,$ $A_{5}\left( x,y\right) ,$ $A_{6}\left( x,y\right) ,$ $B_{1}\left( x,y\right) ,$ $B_{2}\left( x,y\right),$ $B_{3}\left( x,y\right) $ are homogeneous polynomials of degree $a,$ $a,$ $b,$ $b,$ $c,$ $c,$ $n,$ $n,$ $m$ respectively. Concrete example exhibiting the applicability of our result is introduced. |
| format | Article |
| id | doaj-art-293bf051ca06450189a2f6295319608a |
| institution | Kabale University |
| issn | 2544-9990 |
| language | English |
| publishDate | 2021-11-01 |
| publisher | EJAAM |
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| series | E-Journal of Analysis and Applied Mathematics |
| spelling | doaj-art-293bf051ca06450189a2f6295319608a2025-02-08T18:35:22ZengEJAAME-Journal of Analysis and Applied Mathematics2544-99902021-11-01202110.2478/ejaam-2021-0001On the non existence of periodic orbits for a class of two dimensional Kolmogorov systemsRachid Boukoucha0Department of Technology, Faculty of Technology, University of Bejaia, 06000 Bejaia, AlgeriaIn this paper we characterize the integrability and the non-existence of limit cycles of Kolmogorov systems of the form $$ 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.https://ejaam.org/articles/2021/10.2478-ejaam-2021-0001.pdfkolmogorov systemfirst integralperiodic orbitslimit cycle |
| spellingShingle | Rachid Boukoucha On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems E-Journal of Analysis and Applied Mathematics kolmogorov system first integral periodic orbits limit cycle |
| title | On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems |
| title_full | On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems |
| title_fullStr | On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems |
| title_full_unstemmed | On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems |
| title_short | On the non existence of periodic orbits for a class of two dimensional Kolmogorov systems |
| title_sort | on the non existence of periodic orbits for a class of two dimensional kolmogorov systems |
| topic | kolmogorov system first integral periodic orbits limit cycle |
| url | https://ejaam.org/articles/2021/10.2478-ejaam-2021-0001.pdf |
| work_keys_str_mv | AT rachidboukoucha onthenonexistenceofperiodicorbitsforaclassoftwodimensionalkolmogorovsystems |