On the limit cycles for a class of generalized Kukles differential systems |
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Authors: | Amel Boulfoul Amar Makhlouf and Nawal Mellahi |
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Institution: | 20 august 1955 university, Skikda, Algereria.,2.Departement of Mathematics, LMA Laboratory, Badji-Mokhtar University, BP12 El Hadjar, 23000 Annaba, Algeria. and Department of Mathematics, 20 August 1955 University, BP26 El Hadaiek, 21000 Skikda, Algeria |
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Abstract: | In this paper, we consider the limit cycles of a class of polynomial differential systems of the form $\dot{x}=-y, \hspace{0.2cm} \dot{y}=x-f(x)-g(x)y-h(x)y^{2}-l(x)y^{3},$ where $f(x)=\epsilon f_{1}(x)+\epsilon^{2}f_{2}(x),$ $g(x)=\epsilon g_{1}(x)+\epsilon^{2}g_{2}(x),$ $h(x)=\epsilon h_{1}(x)+\epsilon^{2}h_{2}(x)$ and $l(x)=\epsilon l_{1}(x)+\epsilon^{2}l_{2}(x)$ where $f_{k}(x),$ $g_{k}(x),$ $h_{k}(x)$ and $l_{k}(x)$ have degree $n_{1},$ $n_{2},$ $n_{3}$ and $n_{4},$ respectively for each $k=1,2,$ and $\varepsilon$ is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of the linear center $\dot{x}=-y,$ $\dot{y}=x$ using the averaging theory of first and second order. |
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Keywords: | Limit cycle averaging theory Kukles systems |
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