Near-integrability of periodic FPU-chains

The FPU-chain with periodic boundary conditions is studied by Birkhoff–Gustavson normalisation. In the cases of up to 6 particles and for β-chains with an odd number of particles the normal forms are integrable, which permits us to apply KAM-theory. This leads to the presence of many invariant tori on which the motion is quasi-periodic. Thus we explain the recurrence phenomena and the small size of chaos observed in experiments. Furthermore, we find a certain clustering of modes.