Lyapunov instability of a system with long-range forces

We study the largest Lyapunov exponent l and the finite size effects of a system of N fully coupled classical particles, which shows a second order phase transition. Slightly below the critical energy density Uc, l shows a peak which persists for very large N values N 20 000 . We show, both numerically and analytically, that chaoticity is strongly related to kinetic energy fluctuations. In the limit of small energy, l goes to zero with an N-independent power law: l U. In the continuum limit the system is integrable in the whole high temperature phase. More precisely, the behavior l N21 3 is found numerically for U . Uc and justified on the basis of a random matrix approximation.