Physica A 285 (2000) 467– 482 www.elsevier.com/locate/physa Near-integrability of periodic FPU-chains Bob Rink ∗ , Ferdinand Verhulst Mathematisch Instituut, University of Utrecht, PO Box 80.010, 3508 TA Utrecht, The Netherlands Received 9 March 2000 Abstract The FPU-chain with periodic boundary conditions is studied by Birkho–Gustavson normal- isation. In the cases of up to 6 particles and for -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 re- currence phenomena and the small size of chaos observed in experiments. Furthermore, we nd a certain clustering of modes. c  2000 Elsevier Science B.V. All rights reserved. PACS: 63.10.+a; 63.70.+h; 45.10.Hj; 05.45.−a Keywords: Fermi–Pasta–Ulam-chain; Normalisation; KAM-theorem; Recurrence 1. Introduction The FPU-chain with periodic boundary conditions is a model for nonlinear interaction by point masses moving on a circle with nearest-neighbour coupling. In the early 1950s Fermi, Pasta and Ulam numerically integrated chains with xed endpoints, cf. [1–3]. Their expectation that in the presence of nonlinearities, the chain would show stochastic behaviour and thermalisation by energy transport, appeared to be incorrect. Putting initially all the energy in one normal mode, they observed that this energy was shared by only a few other modes, the remaining modes were hardly excited. Additionally, within a considerable time the system returned close to its initial state. On increasing the nonlinearity, this recurrence occurred even earlier. Later computations, e.g. described in Ref. [4], conrmed that these phenomena can also be observed in large periodic chains. Empirical evidence is found that for small total energy, the normal mode energies are hardly shared. Stochastic behaviour can only be observed when the energy level passes a certain critical value. * Corresponding author. Fax: +31-30-2518394. E-mail address: rink@math.uu.nl (B. Rink). 0378-4371/00/$ - see front matter c  2000 Elsevier Science B.V. All rights reserved. PII: S0378-4371(00)00253-3