# Naive question on ODE numerical solving and periodic initial conditions

I have a second-order nonlinear ODE for which I would like to find a numerical solution, with periodic initial (boundary) conditions.

Can this be done directly with NDSolve?

I can of course vary the initial conditions, look at the phase-space (y and y' strobing at the periodicity of my periodic condition) and find fixed points.

Would you be able to suggest a better solution and point me in a direction that would help me?

The equation I consider is $y^{\prime\prime}(s) - \frac{c_1}{y(s)^3} + K(s) y(s) - \frac{c_2}{y(s)} = 0$

where $K(s)$ is periodic with period $T$ and with the boundary conditions $y(0) = y(T)$.

• Now we are getting somewhere. Have you, in any event, looked up Floquet theory? Can your $K(s)$ be expanded as a Fourier series, or is it a more complicated periodic function (e.g. an elliptic function)? – J. M. will be back soon Jun 29 '16 at 13:02
• It can be expanded and that's actually what I'm doing, I'm considering only a few harmonics of the actual $K(s)$. I'm quite OK with Floquet theory and Hills equation but how would you use that with the nonlinear part? – Cedric H. Jun 29 '16 at 13:09
• I am not current with the literature anymore, but Floquet has been extended for some nonlinear ODEs with periodic coefficients; you will want to look up the relevant papers on this. – J. M. will be back soon Jun 29 '16 at 13:11
• OK I'll have a look. But from a pure Mathematica point of view, there is no special trick to force NDSolve to consider periodic solution right? – Cedric H. Jun 29 '16 at 13:13
• If you're looking at NDSolve, then it would be rather helpful to provide at least some specific cases for $K$ and the $c_i$ that you're interested in. – Emilio Pisanty Jun 29 '16 at 13:25