The criticality of centers of potential systems at the outer boundary

The number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied. We call this number the criticality at the outer boundary. Our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1. We apply them to study the bifurcation diagram of the period function of X=−y∂x+((x+1)p−(x+1)q)∂yX=−y∂x+((x+1)p−(x+1)q)∂y with q