<abstract xmlns="http://www.w3.org/1999/xhtml">

We provide sufficient conditions for the existence of periodic solutions for the differential systems 
<disp-formula id="j_amns.2020.2.00079_eq_010">
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2020.2.00079_eq_010.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtable columnalign="center"><mtr columnalign="center"><mtd columnalign="center"><mrow><msup><mi>x</mi><mo>′</mo></msup><mo>=</mo><mi>y</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><msup><mi>y</mi><mo>′</mo></msup><mo>=</mo><mi>z</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><msup><mi>z</mi><mo>′</mo></msup><mo>=</mo><mo>−</mo><mi>y</mi><mo>−</mo><mi>ε</mi><mi>F</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mtext>and</mtext></mrow></mtd></mtr><mtr columnalign="center"><mtd columnalign="center"><mrow><msup><mi>x</mi><mo>′</mo></msup><mo>=</mo><mi>y</mi><mo>,</mo><mo> </mo><msup><mi>y</mi><mo>′</mo></msup><mo>=</mo><mo>−</mo><mi>x</mi><mo>−</mo><mi>ε</mi><mi>G</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo><mo> </mo><msup><mi>z</mi><mo>′</mo></msup><mo>=</mo><mi>u</mi><mo>,</mo><mo> </mo><msup><mi>u</mi><mo>′</mo></msup><mo>=</mo><mo>−</mo><mi>z</mi><mo>−</mo><mi>ε</mi><mi>H</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>u</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd></mtr></mtable></mrow></math>
<tex-math>\matrix{{x' = y,\;\;\;y' = z,\;\;\;z' = - y - \varepsilon F(t,x,y,z),\;\;\;{\rm{and}}} \cr {x' = y,\quad y' = - x - \varepsilon G(t,x,y,z,u),\quad z' = u,\quad u' = - z - \varepsilon H(t,x,y,z,u),} \hfill \cr }</tex-math>
</alternatives>
</disp-formula>
where <italic>F</italic>, <italic>G</italic> and <italic>H</italic> are 2<italic>π</italic>–periodic functions in the variable <italic>t</italic> and <italic>ɛ</italic> is a small parameter. These differential systems appear frequently in many problems coming from the sciences and the engineering.
</abstract>

We provide sufficient conditions for the existence of periodic solutions for the differential systems 



x′=y,   y′=z,   z′=−y−εF(t,x,y,z),   andx′=y, y′=−x−εG(t,x,y,z,u), z′=u, u′=−z−εH(t,x,y,z,u),
\matrix{{x' = y,\;\;\;y' = z,\;\;\;z' = - y - \varepsilon F(t,x,y,z),\;\;\;{\rm{and}}} \cr {x' = y,\quad y' = - x - \varepsilon G(t,x,y,z,u),\quad z' = u,\quad u' = - z - \varepsilon H(t,x,y,z,u),} \hfill \cr }


where F, G and H are 2π–periodic functions in the variable t and ɛ is a small parameter. These differential systems appear frequently in many problems coming from the sciences and the engineering.

Periodic solutions for differential systems in ℝ3 and ℝ4

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

Periodic solutions for differential systems in ℝ and ℝ

We provide sufficient conditions for the existence of periodic solutions for the differential systems...

Periodic solutions for differential systems in ℝ³ and ℝ⁴

Publié en ligne: 31 déc. 2020

Pages: 373 - 380

Reçu: 27 sept. 2019

Accepté: 02 févr. 2020

DOI: https://doi.org/10.2478/amns.2020.2.00079

Mots clés
periodic orbit, third–order differential equation, averaging theory, memory oscillators

© 2020 Amina Feddaoui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Periodic solutions for differential systems in ℝ3 and ℝ4

Publié en ligne: 31 déc. 2020

Pages: 373 - 380

Reçu: 27 sept. 2019

Accepté: 02 févr. 2020

DOI: https://doi.org/10.2478/amns.2020.2.00079

Mots clésperiodic orbit, third–order differential equation, averaging theory, memory oscillators

© 2020 Amina Feddaoui et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Periodic solutions for differential systems in ℝ³ and ℝ⁴

Mots clés
periodic orbit, third–order differential equation, averaging theory, memory oscillators