Find initial condition such that ODE solution is periodic

TomAlso

I have the following ODE system

[itex]
\begin{cases}
x' = v \\
v' = v - \frac{v^3}{3} - x \\
x(0) = x_0 \\
v(0) = 0
\end{cases}
[/itex]

I am asked to find [itex]x_0>0[/itex] such that the solution [itex](x(t),v(t))[/itex] is periodic. Also, I need to find the period [itex]T[/itex] of such solution.

I don't know how to solve the system in the first place (or if it is even possible), so is there a way to figure out what [itex]x_0>0[/itex] will give a periodic solution without solving the system? Thanks!