Upper bounds for the number of limit cycles for a class of polynomial differential systems via the averaging method

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}) \\ \qquad-\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.