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

In this paper, we consider the limit cycles of a class of polynomial differential systems of the form
\begin{equation*} \left\{ \begin{array}{l} \dot{x}=y-\varepsilon (g_{11}\left( x\right) y^{2\alpha +1}+f_{11}\left( x\right) y^{2\alpha })-\varepsilon ^{2}(g_{12}\left( x\right) y^{2\alpha +1}+f_{12}\left( x\right) y^{2\alpha }) ,\\ \dot{y}=-x-\varepsilon (g_{21}\left( x\right) y^{2\alpha +1}+f_{21}\left( x\right) y^{2\alpha })-\varepsilon ^{2}(g_{22}\left( x\right) y^{2\alpha +1}+f_{22}\left( x\right) y^{2\alpha }), \end{array} \right. \end{equation*}
where $m,n,k,l$ and $\alpha $ are positive integers, $g_{1\kappa }$, $ g_{2\kappa },f_{1\kappa }$ and $f_{2\kappa }$ have degree $n,m,l$ and $k$, respectively for each $\kappa =1,2$, 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.