Using the averaging theory of first and second order we study the maximum
number of limit cycles of generalized Liénard differential systems $$\begin{cases} \dot{x}= y + ϵh^1_l
(x) + ϵ^2h^2_l
(x), \\ \dot{y}= −x − ϵ(f^1_n
(x)y^{2p+1}+ g^1_m(x)) + ϵ^2
(f^2_n(x)y^{2p+1}+ g^2_m(x)), \end{cases}$$ which bifurcate from the periodic orbits of the linear center $\dot{x} = y,$ $\dot{y}= −x,$ where $ϵ$ is a small parameter. The polynomials $h^1_l$ and $h^2_l$ have degree $l;$ $f^1_n$ and $f^2_n$ have degree $n;$ and $g^1_m,$ $g^2_m$ have degree $m.$ $p ∈ \mathbb{N}$ and [·] denotes the
integer part function.
