13. Let $f(x)$ be a periodic function defined on $(-\infty,+\infty)$ with a period of 2, and it is an even function. In the interval $[2,3]$, $f(x)=-2(x-3)^{2}+4$. The rectangle $A B C D$ has two vertices $A$ and $B$ on the $x$-axis, and $C$ and $D$ on the graph of the function $y=f(x)(0 \leqslant x \leqslant 2)$. Find the maximum value of the area of this rectangle.
13. When $0 \leqslant x \leqslant 1$, $f(x)=f(x+2)=-2(x-1)^{2}+4$;

When $1 \leqslant x \leqslant 2$, $f(x)=f(-x)=f(-x+4)=-2(-x+1)^{2}+4=-2$ $(x-1)^{2}+4$.
Let $D(x, t)$, $C(2-x, t)$, then $t=-2(x-1)^{2}+4$,
$$
S_{A B B C 7}=|A B| \cdot|B C|=(2-2 x) t=\sqrt{(8-2 t) \cdot t \cdot t} \leqslant \sqrt{\left[\frac{(8-2 t)+t+t}{3}\right]^{3}}=\frac{16 \sqrt{6}}{9} .
$$

When and only when $t=\frac{8}{3}$, i.e., $x=1-\frac{\sqrt{6}}{3}$, the area of quadrilateral $A B C D$ reaches its maximum value $\frac{16 \sqrt{6}}{9}$.