Local chaotic behaviour in the Fermi-Pasta-Ulam system

We discuss the use of the maximal Lyapunov Characteristic Number as a stochasticity indicator in connection with the persistence of the FPU paradox in the thermodynamic limit. We show that the positiveness of the LCN does not imply that the dynamic is ergodic in statistical sense. On the other hand, our numerical exploration suggests that the energy surface may be separated into different chaotic regions that may trap the orbit for a long time. This is compatible with the existence of exponentially long times of relaxation to statistical equilibrium in the sense of Nekhoroshev's theory. Thus, the relevance of the FPU phenomenon for large systems remains a still open problem.