Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory
In this article, we provide sufficient conditions for the existence of periodic solutions for the polynomial differential system of the form x˙=−y+εP1(x,y,z,u,v)+h1(t),y˙=x+εP2(x,y,z,u,v)+h2(t),z˙=−u+εP3(x,y,z,u,v)+h3(t),u˙=z+εP4(x,y,z,u,v)+h4(t),v˙=λv+εP5(x,y,z,u,v)+h5(t),\begin{array}{r}\dot{x}=-y...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2023-12-01
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| Series: | Nonautonomous Dynamical Systems |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/msds-2023-0104 |
| _version_ | 1827398629224087552 |
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| author | Tabet Achref Eddine Makhlouf Amar |
| author_facet | Tabet Achref Eddine Makhlouf Amar |
| author_sort | Tabet Achref Eddine |
| collection | DOAJ |
| description | In this article, we provide sufficient conditions for the existence of periodic solutions for the polynomial differential system of the form x˙=−y+εP1(x,y,z,u,v)+h1(t),y˙=x+εP2(x,y,z,u,v)+h2(t),z˙=−u+εP3(x,y,z,u,v)+h3(t),u˙=z+εP4(x,y,z,u,v)+h4(t),v˙=λv+εP5(x,y,z,u,v)+h5(t),\begin{array}{r}\dot{x}=-y+\varepsilon {P}_{1}\left(x,y,z,u,v)+{h}_{1}\left(t),\\ \dot{y}=x+\varepsilon {P}_{2}\left(x,y,z,u,v)+{h}_{2}\left(t),\\ \dot{z}=-u+\varepsilon {P}_{3}\left(x,y,z,u,v)+{h}_{3}\left(t),\\ \dot{u}=z+\varepsilon {P}_{4}\left(x,y,z,u,v)+{h}_{4}\left(t),\\ \dot{v}=\lambda v+\varepsilon {P}_{5}\left(x,y,z,u,v)+{h}_{5}\left(t),\end{array} where P1,P2,P3,P4{P}_{1},{P}_{2},{P}_{3},{P}_{4}, and P5{P}_{5} are polynomials in the variables x,y,z,u,vx,y,z,u,v of degree nn, hi(t){h}_{i}\left(t) are 2π2\pi -periodic functions with i=1,5¯i=\overline{1,5}, λ\lambda is a real number, and ε\varepsilon is a small parameter. |
| first_indexed | 2024-03-08T19:32:30Z |
| format | Article |
| id | doaj.art-722bf236315d4e658856930612eca934 |
| institution | Directory Open Access Journal |
| issn | 2353-0626 |
| language | English |
| last_indexed | 2024-03-08T19:32:30Z |
| publishDate | 2023-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Nonautonomous Dynamical Systems |
| spelling | doaj.art-722bf236315d4e658856930612eca9342023-12-26T07:40:42ZengDe GruyterNonautonomous Dynamical Systems2353-06262023-12-0110110311110.1515/msds-2023-0104Periodic solutions of some polynomial differential systems in dimension 5 via averaging theoryTabet Achref Eddine0Makhlouf Amar1Department of Mathematics, Laboratory LMA, University of Annaba, El Hadjar, 23000 Annaba, AlgeriaDepartment of Mathematics, Laboratory LMA, University of Annaba, El Hadjar, 23000 Annaba, AlgeriaIn this article, we provide sufficient conditions for the existence of periodic solutions for the polynomial differential system of the form x˙=−y+εP1(x,y,z,u,v)+h1(t),y˙=x+εP2(x,y,z,u,v)+h2(t),z˙=−u+εP3(x,y,z,u,v)+h3(t),u˙=z+εP4(x,y,z,u,v)+h4(t),v˙=λv+εP5(x,y,z,u,v)+h5(t),\begin{array}{r}\dot{x}=-y+\varepsilon {P}_{1}\left(x,y,z,u,v)+{h}_{1}\left(t),\\ \dot{y}=x+\varepsilon {P}_{2}\left(x,y,z,u,v)+{h}_{2}\left(t),\\ \dot{z}=-u+\varepsilon {P}_{3}\left(x,y,z,u,v)+{h}_{3}\left(t),\\ \dot{u}=z+\varepsilon {P}_{4}\left(x,y,z,u,v)+{h}_{4}\left(t),\\ \dot{v}=\lambda v+\varepsilon {P}_{5}\left(x,y,z,u,v)+{h}_{5}\left(t),\end{array} where P1,P2,P3,P4{P}_{1},{P}_{2},{P}_{3},{P}_{4}, and P5{P}_{5} are polynomials in the variables x,y,z,u,vx,y,z,u,v of degree nn, hi(t){h}_{i}\left(t) are 2π2\pi -periodic functions with i=1,5¯i=\overline{1,5}, λ\lambda is a real number, and ε\varepsilon is a small parameter.https://doi.org/10.1515/msds-2023-0104periodic orbitdifferential systemaveraging theory34c2534c2937g15 |
| spellingShingle | Tabet Achref Eddine Makhlouf Amar Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory Nonautonomous Dynamical Systems periodic orbit differential system averaging theory 34c25 34c29 37g15 |
| title | Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory |
| title_full | Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory |
| title_fullStr | Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory |
| title_full_unstemmed | Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory |
| title_short | Periodic solutions of some polynomial differential systems in dimension 5 via averaging theory |
| title_sort | periodic solutions of some polynomial differential systems in dimension 5 via averaging theory |
| topic | periodic orbit differential system averaging theory 34c25 34c29 37g15 |
| url | https://doi.org/10.1515/msds-2023-0104 |
| work_keys_str_mv | AT tabetachrefeddine periodicsolutionsofsomepolynomialdifferentialsystemsindimension5viaaveragingtheory AT makhloufamar periodicsolutionsofsomepolynomialdifferentialsystemsindimension5viaaveragingtheory |