The authors consider the following real planar autonomous piecewise differential system: \[ \begin{array}{lll} \left( \begin{array}{c} \dot{x} \\ \dot{y} \end{array} \right) & = & \left\{ \begin{array}{ll} \displaystyle \left( y + \epsilon f_1^{+}(x,y) + \epsilon^2 f_2^{+}(x,y), -x + \epsilon g_1^{+}(x,y) + \epsilon^2 g_2^{+}(x,y) \right) \text{ for }&y \geq x^m, \vspace{0.2cm} \\ \displaystyle \left( y + \epsilon f_1^{-}(x,y) + \epsilon^2 f_2^{-}(x,y), -x + \epsilon g_1^{-}(x,y) + \epsilon^2 g_2^{-}(x,y) \right)\text{ for } &y < x^m, \end{array} \right\} \end{array} \] where \(0< |\epsilon|<<1\), \(m \in \mathbb{N}\), \(m \geq 1\) and \[ \begin{array}{ll} \displaystyle f_1^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} a_{ij}^{\pm} x^i y^j, & \quad \displaystyle g_1^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} b_{ij}^{\pm} x^i y^j, \vspace{0.2cm} \\ \displaystyle f_2^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} c_{ij}^{\pm} x^i y^j, & \quad \displaystyle g_2^{\pm} (x,y) \, = \, \sum_{i+j=0}^{2} d_{ij}^{\pm} x^i y^j \end{array} \] with \(a_{ij}\), \(b_{ij}\), \(c_{ij}\) and \(d_{ij}\) real parameters. They study the bifurcation of limit cycles from the period annulus corresponding to the value \(\epsilon=0\).

The authors first give the cases where the first Melnikov function is identically null and then, under this assumption, they provide the expression of the second Melnikov function. Denote by \(Z(m)\) the maximum number of limit cycles that bifurcate from the period annulus of the previous system with \(\epsilon=0\) by using up to the second-order Melnikov function. The authors prove the following results:

(i)

If \(m\) is odd, then \(Z(m)=15\);

(ii)

\(Z(2) \, = \, 9\);

(iii)

\(13 \leq Z(4) \leq 19\);

(iv)

\(Z(2k) \geq 14\) for \(k \geq 3\).

These results improve some previously published theorems.