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IMI/Publicaţii/BASM/Ediţii/BASM n3(97), 2021/

Upper Bounds for the Number of Limit Cycles for a Class of Polynomial Differential Systems Via The Averaging Method

Authors: S. Benadouane, A. Berbache, A. Bendjeddou
Keywords: limit cycles, averaging theory, Li\'{e}nard differential systems.

Abstract

In this paper, we study the number of limit cycles of polynomial differential systems of the form% \begin{equation*} \left\{ \begin{array}{l} \dot{x}=y \\ \dot{y}=-x-\varepsilon (h_{1}\left( x\right) y^{2\alpha }+g_{1}\left( x\right) y^{2\alpha +1}+f_{1}\left( x\right) y^{2\alpha +2}) \\ \text{ \ \ \ \ }-\varepsilon ^{2}(h_{2}\left( x\right) y^{2\alpha }+g_{2}\left( x\right) y^{2\alpha +1}+f_{2}\left( x\right) y^{2\alpha +2})% \end{array}% \right. \end{equation*}% where $m,n,k$ and $\alpha $ are positive integers, $h_{i}$, $g_{i}$ and $% f_{i}$ have degree $n,m$ and $k$, respectively for each $i=1,2$, and $% \varepsilon $ is a small parameter. We use the averaging theory of first and second order to provide an accurate upper bound of the number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=y,\dot{y}=-x$. We give an example for which this bound is reached.

Sabah Benadouane
Laboratory of Fundamental and Numerical Mathematics
Department of Mathematics, University Ferhat Abbas S´ etif
E-mail:

Aziza Berbache
Laboratory of Applied Mathematics, University Ferhat
Abbas S´ etif, Department of Mathematics, University of
Bordj Bou Arr´ eridj
E-mail:

Ahmed Bendjeddou
Laboratory of Applied Mathematics, Department of
Mathematics, University Ferhat Abbas S´ etif
E-mail:



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