Maximum Number of Limit Cycles for Generalized Kukles Polynomial Differential Systems

Mellahi, Nawal; Boulfoul, Amel; Makhlouf, Abdenacer

doi:10.1007/s12591-016-0300-3

Cited by 6 publications

(2 citation statements)

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“…where g 1 (x), f 1 (x), g 2 (x) and f 2 (x) have degree k, l, m and n, respectively. • In 2016, using the averaging theory of first and second order, Mellahi et al [25] studied the maximum number of limit cycles bifurcating from the periodic orbits of the linear center,ẋ = −y,ẏ = x perturbed inside a class of generalized Kukles polynomial differential systems…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Limit cycles of Liénard polynomial systems type by averaging method

Boulfoul

Mellahi

2020

Moroccan Journal of Pure and Applied Analysis

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“…where for every k the polynomials l k m (x), f k n 1 (x), k n 2 (x), and h k n 3 (x) have degree m, n 1 , n 2 , and n 3 respectively, d k 0 0 is a real number and ε is a small parameter. This question has been studied in [46] for k = 1, 2, and the authors obtained the following result.…”

Section: A Class Of Generalized Kukles Differential Systemsmentioning

confidence: 96%

Algorithmic Averaging for Studying Periodic Orbits of Planar Differential Systems

Huang¹

2020

Preprint

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One of the main open problems in the qualitative theory of real planar differential systems is the study of limit cycles. In this article, we present an algorithmic approach for detecting how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems via the averaging method. We propose four symbolic algorithms to implement the averaging method. The first algorithm is based on the change of polar coordinates that allows one to transform a considered differential system to the normal form of averaging. The second algorithm is used to derive the solutions of certain differential systems associated to the unperturbed term of the normal of averaging. The third algorithm exploits the partial Bell polynomials and allows one to compute the integral formula of the averaged functions at any order. The last algorithm is based on the aforementioned algorithms and determines the exact expressions of the averaged functions for the considered differential systems. The implementation of our algorithms is discussed and evaluated using several examples. The experimental results have extended the existing relevant results for certain classes of differential systems. CCS CONCEPTS• Computing methodologies → Symbolic and algebraic manipulation; • Symbolic and algebraic algorithms → Symbolic calculus algorithms.

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