A particle of mass m is moving in a potential well, for which the potential energy is given by $\text{U}\left(x\right)={\text{U}}_{\text{0}}\left(1-\cos \text{a}x\right)$ where ${\text{U}}_{\text{0}}$ and are constants. Then(for the small oscillations)                 

$={\text{U}}_{\text{0}}-{\text{U}}_{\text{0}}\cos \text{a}x$

$\text{F}=-\frac{\text{dUn}}{\text{dn}}$

$=\frac{-\text{d}}{\text{dn}}\left[{\text{U}}_{\text{0}}-{\text{U}}_{\text{0}}\cos \text{a}x\right]$

$\text{F}=0+{\text{U}}_{\text{0}}\left[-{\text{s}}_{\max }\right]$

$\text{F}=-{\text{aU}}_{\text{0}}{\text{s}}_{\max }$

For small oscillatory${\text{s}}_{\max }=\text{a}x$

$\text{F}=-{\text{aU}}_{\text{0}}\left[\text{a}x\right]$

$=-{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}\text{n}$

$\text{ma}=-{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}\text{n}$

$\text{a}=\frac{-{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}}{\text{m}}\text{n}$

${\omega }^2=\frac{{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}}{\text{m}}$

$\omega =\sqrt{\frac{{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}}{\text{m}}}$

$\frac{2\pi }{T}=\sqrt{\frac{{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}}{\text{m}}}$

$\text{T}=2\pi \sqrt{\frac{\text{m}}{{\text{a}}^{\text{2}}{\text{U}}_{\text{0}}}}$

$\text{F}=0$ For equilibrium

$\text{So}x=0$