<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

Publié en ligne: 29 mai 2020

Pages: 1 - 15

Reçu: 27 nov. 2019

Accepté: 25 févr. 2020

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

Mots clés
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

Publié en ligne: 29 mai 2020

Pages: 1 - 15

Reçu: 27 nov. 2019

Accepté: 25 févr. 2020

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

Mots clésLimit 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.

Mots clés
Limit cycle, Linard differential system, Periodic orbit, Averaging theory