Coherence angles and coherence times in Hamiltonian systems with many degrees of freedom

The coexistence of ordered and chaotic dynamics in one and the same system has been detected already several years ago. A thorough description of a complex dynamical system, both from a mechanical and from a statistical point of view, requires the determination of the level of order and chaoticity of each single degree of freedom (DOF). We have introduced in a recent paper a new diagnostic tool to analyse the chaoticity of single DOFs or groups of DOFs: the coherence angles, which measure the angular distance between any physically relevant direction and the direction of maximum expansion in the tangent space. They allow at the same time a detailed characterization and a synoptic view of the dynamical behaviour of a system with many DOFs, but lack resolution among the most ordered DOFs when their number is very large. We present here a method allowing the attribution to each DOF (or group of DOFs) of a characteristic coherence time, which overcomes this lack of resolution. In phase space regions characterized by a highly chaotic dynamics, the coherence times are similar. On the other hand, in regions where the dynamics is weakly chaotic, the coherence times show relevant differences in the dynamical behaviour of different groups of DOFs.