How Can I Numerically Calculate the Period of a Nonlinear ODE Trajectory?

[srpvx]

I have a second order nonlinear ODE. I know that a trajectory with specified initial conditions $$\left[ x(0) = x_0, \dot{x}(0) = \dot{x}_0 \right]$$ is periodic. How can I numerically calculate period of this trajectory without solve this DE?

 

[rcgldr]

Are you allowed to use a numerical method like Runge Kutta or would that be considered "solving" the ODE?

 

[srpvx]

Are you allowed to use a numerical method like Runge Kutta or would that be considered "solving" the ODE?

I mean that the numerical solution like the attachment paper.

 

[rcgldr]

That's a method that I'm not familiar with. Hopefully someone here may be able to help.