Limit cycles for a class of generalized Liénard polynomial differential systems via averaging theory

Abstract

For $\left \vert \varepsilon \right \vert $ sufficiently small parameter, we consider the number of limit cycles of the polynomial differential system \begin{equation*} \left \{\begin{array}{l}\dot{x}=y-\sum \limits_{k\geq 1}\varepsilon ^{k}f_{1k}\left( x\right)y^{2\beta }, \\ \dot{y}=-x-\sum \limits_{k\geq 1}\varepsilon ^{k}(f_{2k}\left( x\right) y^{2\beta }+g_{2k}\left( x,y\right) y^{2\alpha +1}),\end{array}\right.\end{equation*} where $g_{2k},f_{1k}$ and $f_{2k}$ are polynomial of, degree $m,n$ and $l$, respectively, for each $k\in \left\{1,2\right \}$ and $\alpha,\beta\in\left \{ 0,1\right\}$. 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 the first and second order.