Variations on the Fermi-Pasta-Ulam chain, a survey

We will present a survey of low energy periodic Fermi-Pasta-Ulam chains with leading idea the "breaking of symmetry". The classical periodic FPU-chain (equal masses for all particles) was analysed by Rink in 2001 with main conclusions that the normal form of the beta-chain is always integrable and that in many cases this also holds for the alfa-chain. The FPU-chain with alternating masses already shows a certain breaking of symmetry. Three exact families of periodic solutions can be identified and a few exact invariant manifolds which are related to the results of Chechin et al.~(1998-2005) on bushes of periodic solutions. An alternating chain of 2n particles is present as submanifold in chains with k 2n particles, k=2, 3, ... . Interaction between the optical and acoustical group in the case of large mass m is demonstrated. The part played by resonance suggests the role of the mass ratios. The 1:1:1:...:1 resonance does not arise for any number of particles and mass ratios. An interesting case is the 1:2:3 resonance that produces after a Hamilton-Hopf bifurcation and breaking symmetry chaotic behaviour in the sense of Shilnikov-Devaney. Another interesting case is the 1:2:4 resonance. As expected the analysis of various cases has a significant impact on recurrence phenomena; this will be illustrated by numerical results.