Periodic solutions for a class of perturbed sixth-order autonomous differential equations

Q2 Mathematics
Chems Eddine Berrehail, A. Makhlouf
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引用次数: 0

Abstract

PurposeThe 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/approachThe 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.FindingsAll 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/valueThe authors studied Equation 1 which depends explicitly on the independent variable t. Here, the authors studied the autonomous case using a different approach.
一类摄动六阶自治微分方程的周期解
目的研究一类六阶自治常微分方程x(6)+(1+p2+q2)x的周期解+(p2+q2+p2q2)x¨+p2q2x=εF(x,ẋ,x¨,x…,x……,x(5)),其中p和q是不同于1,0,−1的有理数,p≠q,ε是一个足够小的参数,F∈C2是一个非线性自治函数。设计/方法/方法作者应使用平均理论来研究一类扰动六阶自治微分方程(DE)的周期解。平均理论是研究具有周期强迫的非线性微分系统动力学的经典工具。平均理论有着悠久的历史,始于拉格朗日和拉普拉斯的经典著作。平均理论被用于研究二阶和更高阶DE的周期解。定理1给出了一类扰动六阶自治DE周期解的主要结果。作者提出了一些应用来说明主要结果。原创性/价值作者研究了明确依赖于自变量t的方程1。在这里,作者使用不同的方法研究了自主情况。
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来源期刊
Arab Journal of Mathematical Sciences
Arab Journal of Mathematical Sciences Mathematics-Mathematics (all)
CiteScore
1.20
自引率
0.00%
发文量
17
审稿时长
8 weeks
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