  We consider the family of dehomogenized Loud's centers
$X_{\mu}=y(x-1)\partial_x+(x+Dx^2+Fy^2)\partial_y,$ where
$\mu=(D,F)\in\mathbb{R}^2,$ and we study the number of critical periodic orbits
that emerge or dissapear from the polycycle at the boundary of the period
annulus. This number is defined exactly the same way as the well-known notion
of cyclicity of a limit periodic set and we call it criticality. The previous
results on the issue for the family $\{X_{\mu},\mu\in\mathbb{R}^2\}$
distinguish between parameters with criticality equal to zero (regular
parameters) and those with criticality greater than zero (bifurcation
parameters). A challenging problem not tackled so far is the computation of the
criticality of the bifurcation parameters, which form a set $\Gamma_{B}$ of
codimension 1 in $\mathbb{R}^2$. In the present paper we succeed in proving
that a subset of $\Gamma_{B}$ has criticality equal to one.
