<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0005_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow></msup><mo>-</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>λ</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>μ</mi></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow></msup><mo>+</mo><mi>A</mi><mover><mi>x</mi><mo>⃜</mo></mover><mo>+</mo><mi>B</mi><mover><mi>x</mi><mo>¨</mo></mover><mo>+</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><msup><mrow><mi>λ</mi></mrow><mn>2</mn></msup><msup><mrow><mi>μ</mi></mrow><mn>2</mn></msup><mi>x</mi><mo>=</mo><mi>ɛ</mi><mi>F</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mover><mi>x</mi><mo>˙</mo></mover><mo>,</mo><mover><mi>x</mi><mo>¨</mo></mover><mo>,</mo><mover><mi>x</mi><mo>⃛</mo></mover><mo>,</mo><mover><mi>x</mi><mo>⃜</mo></mover><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow></msup><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow></msup></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math><tex-math>{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),</tex-math></alternatives></disp-formula>
where <italic>A</italic> = <italic>p</italic>2λ2 + <italic>p</italic>2<italic>µ</italic>2 + λ2<italic>µ</italic>2 + <italic>p</italic>2 + λ2 + <italic>µ</italic>2, <italic>B</italic> = <italic>p</italic>2 λ2 + <italic>p</italic>2<italic>µ</italic>2 + λ2<italic>µ</italic>2 + <italic>p</italic>2λ2<italic>µ</italic>2, with λ, <italic>µ</italic> and <italic>p</italic> are rational numbers different from −1, 0, 1, and <italic>p</italic> ≠ ±λ, <italic>p</italic> ≠±<italic>µ</italic>, λ ≠±<italic>µ</italic>, ɛ is sufficiently small and <italic>F</italic> is a nonlinear non-autonomous periodic function. Moreover we provide some applications.</abstract>

In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation
x(8)-(1+p2+λ2+μ2)x(6)+Ax⃜+Bx¨+p2λ2μ2x=ɛF(t,x,x˙,x¨,x⃛,x⃜,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.

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

Moroccan Journal of Pure and Applied Analysis

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"On the limit cycles for a class of eighth-order differential equations"}

In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential...

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

Online veröffentlicht: 29. Mai 2020

Seitenbereich: 53 - 61

Eingereicht: 13. Feb. 2020

Akzeptiert: 09. Apr. 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0005

Schlüsselwörter
Periodic orbit, Eighth-order differential equation, Averaging theory

© 2020 Chems Eddine Berrehail et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Online veröffentlicht: 29. Mai 2020

Seitenbereich: 53 - 61

Eingereicht: 13. Feb. 2020

Akzeptiert: 09. Apr. 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0005

SchlüsselwörterPeriodic orbit, Eighth-order differential equation, Averaging theory

© 2020 Chems Eddine Berrehail et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Schlüsselwörter
Periodic orbit, Eighth-order differential equation, Averaging theory