Electron. J. Differential Equations, Vol. 2016 (2016), No. 318, pp. 1-22.

Upper bounds for the number of limit cycles of polynomial differential system

Selma Ellaggoune, Sabrina Badi

Abstract:
For $\varepsilon$ small we consider the number of limit cycles of the polynomial differential system
$$ \dot{x}=y-f_1(x,y)y, \quad \dot{y}=-x-g_2(x,y)-f_2(x,y)y, $$
where $f_1(x,y)=\varepsilon f_{11}(x,y)+\varepsilon^2f_{12}(x,y)$, $g_2(x,y)=\varepsilon g_{21}(x,y)+\varepsilon^2 g_{22}(x,y)$ and $f_2(x,y)=\varepsilon f_{21}(x,y)+\varepsilon^2 f_{22}(x,y)$ where $f_{1i}, f_{2i}, g_{2i}$ have degree $l, n,m$ respectively for each $i=1,2$. We provide an accurate upper bound of the maximum number of limit cycles that this class of systems can have bifurcating from the periodic orbits of the linear center $\dot{x}=y, \dot{y}=-x$ using the averaging theory of first and second order. We give an example for which this bound is reached.

Submitted October 1, 2016. Published December 14, 2016.
Math Subject Classifications: 34C25, 34C29, 37G15.
Key Words: Limit cycle; Lienard differential equation; averaging method.

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Selma Ellaggoune
Laboratory of Applied Mathematics and Modeling
University 8 Mai 1945
Guelma, Algeria
email: sellaggoune@gmail.com
Sabrina Badi
Laboratory of Applied Mathematics and Modeling
University 8 Mai 1945
Guelma, Algeria
email: badisabrina@yahoo.fr

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