I'm working on a parameter study of Duffing's equation
$\ddot x + \delta \dot x + \alpha x + \beta x^3 = \gamma \cos{\omega t},$
where $\delta, \alpha, \beta, \gamma$ and $\omega$ are real parameters and $\dot x := dx/dt$. Substituting $x \rightarrow u, \dot x\rightarrow v$ in the usual manner, the idea is to integrate the system
$\dot u = v$,
$\dot v = \gamma \cos{\omega t} - \delta v - \alpha u - \beta u^3$
using the fourth order Runge-Kutta method. I've chosen this pretty much because it's standard. My questions are:
1) How do I determine whether this method is at all applicable to the problem? Is the problem stiff, and should I choose a different method?
This of course is complicated by the fact that Duffing's equation doesn't have an analytical solution for most parameters.
2) How do I determine what step size to use - ie. how is the error resulting from a given step size estimated? (I would very much like to avoid implementing an adaptive step size.)
Being pointed towrads good references will be much appreciated! Best regards, \T