Periodic solutions for a class of perturbed sixth-order autonomous differential equations
Purpose – The objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough pa...
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| Language: | English |
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Emerald Publishing
2025-01-01
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| Series: | Arab Journal of Mathematical Sciences |
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| Online Access: | https://www.emerald.com/insight/content/doi/10.1108/AJMS-02-2022-0045/full/pdf |
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| author | Chems Eddine Berrehail Amar Makhlouf |
| author_facet | Chems Eddine Berrehail Amar Makhlouf |
| author_sort | Chems Eddine Berrehail |
| collection | DOAJ |
| description | Purpose – The objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough parameter and F ∈ C2 is a nonlinear autonomous function. Design/methodology/approach – The authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs. Findings – All the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results. Originality/value – The authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach. |
| format | Article |
| id | doaj-art-44d764ebe6184e1cbed2ab15deacd6b5 |
| institution | Kabale University |
| issn | 1319-5166 2588-9214 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Emerald Publishing |
| record_format | Article |
| series | Arab Journal of Mathematical Sciences |
| spelling | doaj-art-44d764ebe6184e1cbed2ab15deacd6b52025-01-24T04:22:17ZengEmerald PublishingArab Journal of Mathematical Sciences1319-51662588-92142025-01-01311223210.1108/AJMS-02-2022-0045Periodic solutions for a class of perturbed sixth-order autonomous differential equationsChems Eddine Berrehail0Amar Makhlouf1Department of Mathematics, Faculty of Sciences, University of Badji Mokhtar Annaba, Annaba, AlgeriaDepartment of Mathematics, Faculty of Sciences, University of Badji Mokhtar Annaba, Annaba, AlgeriaPurpose – The objective of this work is to study the periodic solutions for a class of sixth-order autonomous ordinary differential equations x(6)+(1+p2+q2)x… .+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x… .,x(5)), where p and q are rational numbers different from 1, 0, −1 and p ≠ q, ε is a small enough parameter and F ∈ C2 is a nonlinear autonomous function. Design/methodology/approach – The authors shall use the averaging theory to study the periodic solutions for a class of perturbed sixth-order autonomous differential equations (DEs). The averaging theory is a classical tool for the study of the dynamics of nonlinear differential systems with periodic forcing. The averaging theory has a long history that begins with the classical work of Lagrange and Laplace. The averaging theory is used to the study of periodic solutions for second and higher order DEs. Findings – All the main results for the periodic solutions for a class of perturbed sixth-order autonomous DEs are presenting in the Theorem 1. The authors present some applications to illustrate the main results. Originality/value – The authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach.https://www.emerald.com/insight/content/doi/10.1108/AJMS-02-2022-0045/full/pdfPeriodic orbitSixth-order differential equationAveraging theory |
| spellingShingle | Chems Eddine Berrehail Amar Makhlouf Periodic solutions for a class of perturbed sixth-order autonomous differential equations Arab Journal of Mathematical Sciences Periodic orbit Sixth-order differential equation Averaging theory |
| title | Periodic solutions for a class of perturbed sixth-order autonomous differential equations |
| title_full | Periodic solutions for a class of perturbed sixth-order autonomous differential equations |
| title_fullStr | Periodic solutions for a class of perturbed sixth-order autonomous differential equations |
| title_full_unstemmed | Periodic solutions for a class of perturbed sixth-order autonomous differential equations |
| title_short | Periodic solutions for a class of perturbed sixth-order autonomous differential equations |
| title_sort | periodic solutions for a class of perturbed sixth order autonomous differential equations |
| topic | Periodic orbit Sixth-order differential equation Averaging theory |
| url | https://www.emerald.com/insight/content/doi/10.1108/AJMS-02-2022-0045/full/pdf |
| work_keys_str_mv | AT chemseddineberrehail periodicsolutionsforaclassofperturbedsixthorderautonomousdifferentialequations AT amarmakhlouf periodicsolutionsforaclassofperturbedsixthorderautonomousdifferentialequations |