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This paper deals with the period function of the reversible quadratic centers\begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y,\end{equation*} where $\np=(D,F)\in\R^2.$ Compactifying the vector field to$\Sc^2$, the boundary of the period annulus has two connected components, thecenter itself and a polycycle. We call them the inner and outer boundary of theperiod annulus, respectively. We are interested in the bifurcation of criticalperiodic orbits from the polycycle $\out_\np$ at the outer boundary. A criticalperiod is an isolated critical point of the period function. The criticality ofthe period function at the outer boundary is the maximal number of criticalperiodic orbits of $X_\np$ that tend to $\out_{\np_0}$ in the Hausdorff senseas $\np\to\np_0.$ This notion is akin to the cyclicity in Hilbert's 16thProblem. Our main result (Theorem A) shows that the criticality at the outerboundary is at most 2 for all $\np=(D,F)\in\R^2$ outside the segments$\{-1\}\times [0,1]$ and $\{0\}\times [0,2]$. With regard to the bifurcationfrom the inner boundary, Chicone and Jacobs proved in their seminal paper onthe issue that the upper bound is 2 for all $\np\in\R^2.$ In this paper thetechniques are different because, while the period function extendsanalytically to the center, it has no smooth extension to the polycycle. Weshow that the period function has an asymptotic expansion near the polycyclewith the remainder being uniformly flat with respect to~$\np$ and where theprincipal part is given in a monomial scale containing a deformation of thelogarithm. More precisely, Theorem~A follows by obtaining the asymptoticexpansion to fourth order and computing its coefficients, which are notpolynomial in~$\np$ but transcendental.

The criticality of reversible quadratic centers at the outer boundary of its period annulus