Correlations in area preserving maps: a Shannon entropy approach

SHEVCHENKO, IVAN I.; CINCOTTA, PABLO M

PHYSICA D - NONLINEAR PHENOMENA

ELSEVIER SCIENCE BV

Lugar: Amsterdam; Año: 2020

0167-2789

In the present work we extend and generalize the formulation of the Shannonentropy 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 statevariables. The formulation given here includes the analysis of the evolutionof a single variable of the system, for instance a given phase; the phase spacevariables of a 2-dimensional model or the action space of a 4-dimensional mapor a 3dof Hamiltonian. We show that the Shannon entropy provides a directmeasure of the volume of the phase space occupied by a given trajectory aswell as a direct measure of the correlations among the successive values ofthe phase space variables in any dynamical system, in particular when themotion is highly chaotic. We use the standard map model at large valuesof the perturbation parameter to confront all the analytical estimates withthe numerical simulations. The numerical-experimental results show the efficiency of the entropy in revealing the fine structure of the phase space, inparticular the existence of small stability domains (islands around periodicsolutions) that affect the diffusion.