Find initial condition such that ODE solution is periodic

  1. 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!
     
  2. jcsd
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