Correlations in area preserving maps: a Shannon entropy approach

In the present work we extend and generalize the formulation of the Shannon entropy as a measure of correlations in the phase space variables of any dynamical system. By means of theoretical arguments we show that the Shannon entropy is a quite sensitive approach to detect correlations in the state variables. The formulation given here includes the analysis of the evolution of a single variable of the system, for instance a given phase; the phase space variables of a 2-dimensional model or the action space of a 4-dimensional map or a 3dof Hamiltonian. We show that the Shannon entropy provides a direct measure of the volume of the phase space occupied by a given trajectory as well as a direct measure of the correlations among the successive values of the phase space variables in any dynamical system, in particular when the motion is highly chaotic. We use the standard map model at large values of the perturbation parameter to confront all the analytical estimates with the numerical simulations. The numerical-experimental results show the efficiency of the entropy in revealing the fine structure of the phase space, in particular the existence of small stability domains (islands around periodic solutions) that affect the diffusion.