A Variational Principle for Eigenvalue Problems of Hamiltonian Systems

Abstract

We consider the bifurcation problem $u'' + λu = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $λ$. We give a variational characterization of the bifurcating branch $λ$ 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.