Limit cycles of the sixth-order non-autonomous differential equation

PurposeThis study aims to provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.Design/methodology/approachThe authors shall use the averaging theory, more precisely Theorem $6$.FindingsThe main results on the periodic solutions of the fifth-order differential equation (equation (1)) are given in the statement of Theorem 1 and 2.Originality/valueIn this article, the authors provide sufficient conditions for the existence of periodic solutions of the fifth-order differential equation.

show abstract

“…In Refs. [7][8][9], the authors studied the limit cycles of the fourth-, sixth-and eighth-order non-autonomous differential equations.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Periodic solutions for a class of fifth-order differential equations

Berrehail

Bouslah

2022

AJMS

View full text Add to dashboard Cite

show abstract

“…We denote by x(z, t, «) the solution of the differential system (9) such that x(z, 0, «) 5 z. We assume that the unperturbed system _ x ¼ F 0 ðx; tÞ; (10) has an open set V with Cl(V) ⊂ Ω such that for each z ∈ Cl(V), x(t, z, 0) is T-periodic.…”

Section: Averaging Theorymentioning

confidence: 99%

“…In [10], the authors studied the periodic solution of the following sixth-order differential equation:…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Berrehail

Makhlouf

2022

AJMS

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Periodic Solutions for Differential Systems in $\mathbb {R}^{5}$ and $\mathbb {R}^{6}$

Rabia

Makhlouf

2022

J Dyn Control Syst

View full text Add to dashboard Cite

Limit cycles of the sixth-order non-autonomous differential equation

Cited by 4 publications

References 7 publications

Periodic solutions for a class of fifth-order differential equations

Periodic solutions for a class of fifth-order differential equations

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

Periodic Solutions for Differential Systems in $\mathbb {R}^{5}$ and $\mathbb {R}^{6}$

Contact Info

Product

Resources

About