# Find initial condition such that ODE solution is periodic

1. Mar 16, 2012

### TomAlso

I have the following ODE system

$\begin{cases} x' = v \\ v' = v - \frac{v^3}{3} - x \\ x(0) = x_0 \\ v(0) = 0 \end{cases}$

I am asked to find $x_0>0$ such that the solution $(x(t),v(t))$ is periodic. Also, I need to find the period $T$ 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 $x_0>0$ will give a periodic solution without solving the system? Thanks!