Correlations in area preserving maps: A Shannon entropy approach

P. M. Cincotta, I. I. Shevchenko

Research output

Abstract

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 herein 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.

Original languageEnglish
Article number132235
JournalPhysica D: Nonlinear Phenomena
DOIs
Publication statusAccepted/In press - 2019

Fingerprint

Shannon Entropy
preserving
Phase Space
Entropy
entropy
Dynamical systems
Hamiltonians
dynamical systems
Dynamical system
Standard Map
Formulation
Parameter Perturbation
Fine Structure
formulations
Trajectories
Periodic Solution
Trajectory
Computer simulation
Numerical Simulation
fine structure

Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Cite this

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abstract = "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 herein 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.",
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