13. Let $f(x)$ be an odd function with a period of 2, and $f\left(-\frac{2}{5}\right)=3$. If $\sin \alpha=\frac{\sqrt{5}}{5}$, then the value of $f(4 \cos 2 \alpha)$ is
13. -3.

Since $\sin \alpha=\frac{\sqrt{5}}{5}$, therefore,
$$
\cos 2 \alpha=1-2 \sin ^{2} \alpha=\frac{3}{5} \text{. }
$$

Also, $f(x)$ is an odd function with a period of 2, and $f\left(-\frac{2}{5}\right)=$
3, so,
$$
f(4 \cos 2 \alpha)=f\left(\frac{12}{5}\right)=f\left(\frac{2}{5}\right)=-f\left(-\frac{2}{5}\right)=-3 .
$$