Solution
= Solution
For $|z|\ll a$, <linearization> gives
$$
\ddot z=-\frac{2mk}{a^3}z+O(z^3).
$$
The leading motion is <simple harmonic motion> with angular frequency
$$
\omega_z=\sqrt{\frac{2mk}{a^3}}.
$$
With $z(0)=z_0$ and $\dot z(0)=0$,
$$
z(t)\simeq z_0\cos(\omega_zt),
$$
and its period is
$$
\boxed{T=2\pi\sqrt{\frac{a^3}{2mk}}}.
$$