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

<p>We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre ̇<italic>x</italic> = <italic>y</italic>, ̇<italic>y</italic> = −<italic>x</italic>, when it is perturbed in the form
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amns.2021.2.00180_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mover accent="true"><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi><mo>-</mo><mi>ɛ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>cos</mo></mrow></mrow><mi>l</mi></msup><mi>θ</mi></mrow><mo>)</mo></mrow><mi>P</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mover accent="true"><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mi>x</mi><mo>-</mo><mi>ɛ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>cos</mo></mrow></mrow><mi>m</mi></msup><mi>θ</mi></mrow><mo>)</mo></mrow><mi>Q</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math>
<tex-math>\dot x = y - \varepsilon \left( {1 + {{\cos }^l}\theta } \right)P\left( {x,y} \right),\,\,\,\,\dot y =  - x - \varepsilon \left( {1 + {{\cos }^m}\theta } \right)Q\left( {x,y} \right),</tex-math>
</alternatives>
</disp-formula>
where <italic>ε</italic> > 0 is a small parameter, <italic>l</italic> and <italic>m</italic> are positive integers, <italic>P</italic>(<italic>x</italic>, <italic>y</italic>) and <italic>Q</italic>(<italic>x</italic>, <italic>y</italic>) are arbitrary polynomials of degree <italic>n</italic>, and <italic>θ</italic> = arctan(<italic>y</italic>/<italic>x</italic>). As we shall see the differential system (1) is a generalisation of the Mathieu differential equation. The tool for studying such limit cycles is the averaging theory.</p>
</abstract>

We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre ̇x = y, ̇y = −x, when it is perturbed in the form



x˙=y-ɛ(1+coslθ)P(x,y),    y˙=-x-ɛ(1+cosmθ)Q(x,y),
\dot x = y - \varepsilon \left( {1 + {{\cos }^l}\theta } \right)P\left( {x,y} \right),\,\,\,\,\dot y =  - x - \varepsilon \left( {1 + {{\cos }^m}\theta } \right)Q\left( {x,y} \right),


where ε > 0 is a small parameter, l and m are positive integers, P(x, y) and Q(x, y) are arbitrary polynomials of degree n, and θ = arctan(y/x). As we shall see the differential system (1) is a generalisation of the Mathieu differential equation. The tool for studying such limit cycles is the averaging theory.

Limit cycles of a generalised Mathieu differential system

Department of Mathematics and Computer Science

Department of Applied Mathematics and Statistics

Department of Mathematics, Faculty of Sciences

Applied Mathematics and Nonlinear Sciences

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

{"article-title":"Limit cycles of a generalised Mathieu differential system"}

We study the maximum number of limit cycles which bifurcate from the periodic orbits of the linear centre ̇x = y, ̇y = −x, when it is perturbed in the...

Limit cycles of a generalised Mathieu differential system

Published Online: Apr 06, 2024

Page range: -

Received: Jun 17, 2021

Accepted: Sep 24, 2021

DOI: https://doi.org/10.2478/amns.2021.2.00180

Keywords
Limit cycle, averaging theory, differential system

© 2024 Zouhair Diab et al., published by Sciendo

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

Limit cycles of a generalised Mathieu differential system

Published Online: Apr 06, 2024

Page range: -

Received: Jun 17, 2021

Accepted: Sep 24, 2021

DOI: https://doi.org/10.2478/amns.2021.2.00180

KeywordsLimit cycle, averaging theory, differential system

© 2024 Zouhair Diab et al., published by Sciendo

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

Keywords
Limit cycle, averaging theory, differential system