We consider the symmetric 8-loop Hamiltonian potential given by \( H(x,y) = y^2/2 + (x-1)^2(x+1)^2 \) and examine a piecewise polynomial potential perturbation. We begin by obtaining the first-order Melnikov functions \( M_1, M_2, M_3 \) , defined near the three period annuli located at the center points \( (-1,0)\) , (1, 0), and the saddle point at (0, 0), respectively. Upper bounds for the number of simple zeros of these Melnikov functions are provided both individually and simultaneously in terms of the degree n. These simple zeros correspond to limit cycles bifurcating from the three period annuli in various configurations. For low degrees \( n = 2, 3, 4, 5 \) , we explicitly present all possible simultaneous configurations of limit cycles that can bifurcate. Due to the large number of cases and computational difficulties, for degrees \( 6 \le n \le 21 \) , we explore the potential configurations of limit cycles bifurcating from each period annulus, proving the existence of the configuration that we believe to be maximal. Bifurcation diagrams estimating the size of the regions in the parameter space where the best configuration of limit cycles exists are also presented.
