  We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point
boundary conditions where $N(u)$ is a general nonlinear term which may also
depend on the eigenvalue $\lambda$. We give a variational characterization of
the bifurcating branch $\lambda$ as a function of the amplitude of the
solution. As an application we show how it can be used to obtain simple
approximate closed formulae for the period of large amplitude oscillations.
