On the Limit Cycles for a Class of Generalized Kukles Differential Systems

Boulfoul, Amel; Makhlouf, Abdenacer; Mellahi, Nawal

doi:10.11948/2156-907x.20180083

Cited by 3 publications

(9 citation statements)

References 19 publications

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“…Boulfoul and Makhlouf 18,19 studied a more general Lienard system than (1). Also, by using the averaging theory of first and second order, Boulfoul et al 20 studied a class of Kukles systems of the form…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Expression ( 7) is obtained by comparing each term in 𝜆 to all terms in 𝜆 3 and taking into account that 𝜇 0 = 𝜇 or 𝜇 0 = 𝜇 1 or 𝜇 0 = 𝜇 ′ . • Also, it has been shown in Boulfoul et al 20 that there exists generalized Kukles differential equations ( 4), having at least…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Also, by using the averaging theory of first and second order, Boulfoul et al 20 studied a class of Kukles systems of the form

\begin{matrix} right \dot{x} & left = y \\ right \dot{y} & left = - x - f (x) - g (x) y - h (x) y^{2} - k (x) y^{3}, \end{matrix}

where f ( x ), g ( x ), h ( x ) and k ( x ) are polynomials of given degree.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Motivated by the results of previous papers 15,20 and by the increased interest of the recherche on Lienard and Kukles systems, using the averaging theory, we want to study the maximum number of limit cycles which can bifurcate from the periodic orbits of the linear center perturbed inside the class of generalized polynomial differential systems

\begin{matrix} right \dot{x} = & left y - ε (g_{11} (x) + f_{11} (x) y) - ε^{2} (g_{12} (x) + f_{12} (x) y), \\ right \dot{y} = & left - x - ε (g_{21} (x) + f_{21} (x) y - p_{1} (x) y^{2} - q_{1} (x) y^{3}) \\ right & left - ε^{2} (g_{22} (x) + f_{22} (x) y - p_{2} (x) y^{2} - q_{2} (x) y^{3}), \end{matrix}

where f 1 i , f 2 i , g 1 i , g 2 i , p i , and q i have degree l , n , k , m , n 1 , and n 2 , respectively, for each

i = 1,2

. Furthermore, ε is a small parameter.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We consider system (5), where We have that F 10 be identically zero, and with an easy computation, we obtain This equation has exactly three positive zeros. In Boulfoul et al, 20 this system when g 1, i ≡ f 1, i ≡ 0 for each i = 1, 2 has at most two positive zeros using the averaging theory of second order.…”

Section: Lemma 4 the Integral H(r) Is A Polynomial In The Variable R Given Bymentioning

confidence: 99%

See 4 more Smart Citations

Limit cycles of a class of planar polynomial differential systems

Debz

Boulfoul

Berkane

2021

Math Methods in App Sciences

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Also, by using the averaging theory of first and second order, Boulfoul et al 20 studied a class of Kukles systems of the form

\begin{matrix} right \dot{x} & left = y \\ right \dot{y} & left = - x - f (x) - g (x) y - h (x) y^{2} - k (x) y^{3}, \end{matrix}

where f ( x ), g ( x ), h ( x ) and k ( x ) are polynomials of given degree.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

\begin{matrix} right \dot{x} = & left y - ε (g_{11} (x) + f_{11} (x) y) - ε^{2} (g_{12} (x) + f_{12} (x) y), \\ right \dot{y} = & left - x - ε (g_{21} (x) + f_{21} (x) y - p_{1} (x) y^{2} - q_{1} (x) y^{3}) \\ right & left - ε^{2} (g_{22} (x) + f_{22} (x) y - p_{2} (x) y^{2} - q_{2} (x) y^{3}), \end{matrix}

where f 1 i , f 2 i , g 1 i , g 2 i , p i , and q i have degree l , n , k , m , n 1 , and n 2 , respectively, for each

i = 1,2

. Furthermore, ε is a small parameter.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Lemma 4 the Integral H(r) Is A Polynomial In The Variable R Given Bymentioning

confidence: 99%

See 3 more Smart Citations

Limit cycles of a class of planar polynomial differential systems

Debz

Boulfoul

Berkane

2021

Math Methods in App Sciences

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Maximum number of limit cycles for generalized Kukles differential system

Melki

Makhlouf

2022

Journal of Applied Analysis

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We apply the averaging theory of first and second order to a class of generalized polynomial Kukles differential systems, which can bifurcate from the periodic orbits of the linear center x ˙ = y {\dot{x}=y} , y ˙ = - x {\dot{y}=-x} , in order to study the maximum number of limit cycles of these systems.

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Limit cycles of Liénard polynomial systems type by averaging method

Boulfoul

Mellahi

2020

Moroccan Journal of Pure and Applied Analysis

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

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On the Limit Cycles for a Class of Generalized Kukles Differential Systems

Cited by 3 publications

References 19 publications

Limit cycles of a class of planar polynomial differential systems

Limit cycles of a class of planar polynomial differential systems

Maximum number of limit cycles for generalized Kukles differential system

Limit cycles of Liénard polynomial systems type by averaging method

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