On the limit cycles for a class of eighth-order differential equations

Berrehail, Chems Eddine; Bouslah, Zineb; Makhlouf, Amar

doi:10.2478/mjpaa-2020-0005

Abstract

In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation $x^{(8)} - (1 + p^{2} + λ^{2} + μ^{2}) x^{(6)} + A \overset{⃜}{x} + B \ddot{x} + p^{2} λ^{2} μ^{2} x = ɛ F (t, x, \dot{x}, \ddot{x}, \overset{⃛}{x}, \overset{⃜}{x}, x^{(5)}, x^{(6)} x^{(7)}),$ {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p²λ² + p²µ² + λ²µ² + p² + λ² + µ², B = p² λ² + p²µ² + λ²µ² + p²λ²µ², with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

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On the limit cycles for a class of eighth-order differential equations

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