<abstract xmlns="http://www.w3.org/1999/xhtml">We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0001_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>y</mi><mo>,</mo><mi> </mi><mi> </mi><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mi>x</mi><mo>-</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>y</mi><mo>-</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mi>y</mi></mrow><mn>2</mn></msup><mo>,</mo></mrow></math><tex-math>\dot x = y - 1\left( x \right)y,\,\,\dot y = - x - f\left( x \right) - g\left( x \right)y - h\left( x \right){y^2},</tex-math></alternatives></disp-formula>where <italic>l</italic>(<italic>x</italic>) = <italic>∊l</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>l</italic>2(<italic>x</italic>), <italic>f</italic> (<italic>x</italic>) = <italic>∊ f</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>f</italic>2(<italic>x</italic>), <italic>g</italic>(<italic>x</italic>) = <italic>∊g</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>g</italic>2(<italic>x</italic>) and <italic>h</italic>(<italic>x</italic>) = <italic>∊h</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>h</italic>2(<italic>x</italic>) where <italic>lk</italic>(<italic>x</italic>) has degree <italic>m</italic> and <italic>fk</italic>(<italic>x</italic>), <italic>gk</italic>(<italic>x</italic>) and <italic>hk</italic>(<italic>x</italic>) have degree <italic>n</italic> for each <italic>k</italic> = 1, 2, and <italic>∊</italic> is a small parameter.</abstract>

We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the formx˙=y-1(x)y,  y˙=-x-f(x)-g(x)y-h(x)y2,\dot x = y - 1\left( x \right)y,\,\,\dot y =  - x - f\left( x \right) - g\left( x \right)y - h\left( x \right){y^2},where l(x) = ∊l1(x) + ∊2l2(x), f (x) = ∊ f1(x) + ∊2f2(x), g(x) = ∊g1(x) + ∊2g2(x) and h(x) = ∊h1(x) + ∊2h2(x) where lk(x) has degree m and fk(x), gk(x) and hk(x) have degree n for each k = 1, 2, and ∊ is a small parameter.

Limit cycles of Liénard polynomial systems type by averaging method

Moroccan Journal of Pure and Applied Analysis

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

{"article-title":"Limit cycles of Liénard polynomial systems type by averaging method"}

We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the...

Limit cycles of Liénard polynomial systems type by averaging method

Published Online: May 29, 2020

Page range: 1 - 15

Received: Nov 27, 2019

Accepted: Feb 25, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0001

Keywords
Limit cycle, Linard differential system, Periodic orbit, Averaging theory

© 2020 Amel Boulfoul et al., published by Sciendo

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

Limit cycles of Liénard polynomial systems type by averaging method

Published Online: May 29, 2020

Page range: 1 - 15

Received: Nov 27, 2019

Accepted: Feb 25, 2020

DOI: https://doi.org/10.2478/mjpaa-2020-0001

KeywordsLimit cycle, Linard differential system, Periodic orbit, Averaging theory

© 2020 Amel Boulfoul et al., published by Sciendo

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

Keywords
Limit cycle, Linard differential system, Periodic orbit, Averaging theory