Asymptotic resonance, interaction of modes and subharmonic bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam beta-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.