Statistical characterization of the standard map. (15th June 2017) - Ruiz, Guiomar
Abstract: The standard map, paradigmatic conservative system in the ( x, p ) phase space, has been recently shown (Tirnakli and Borges (2016 Sci. Rep .6 23644)) to exhibit interesting statistical behaviors directly related to the value of the standard map external parameter K . A comprehensive statistical numerical description is achieved in the present paper. More precisely, for large values of K (e.g. K = 10) where the Lyapunov exponents are neatly positive over virtually the entire phase space consistently with Boltzmann–Gibbs (BG) statistics, we verify that the q -generalized indices related to the entropy production q e n t, the sensitivity to initial conditions q s e n, the distribution of a time-averaged (over successive iterations) phase-space coordinate q s t a t, and the relaxation to the equilibrium final state q r e l, collapse onto a fixed point, i.e. q e n t = q s e n = q s t a t = q r e l = 1 . In remarkable contrast, for small values of K (e.g. K = 0.2) where the Lyapunov exponents are virtually zero over the entire phase space, we verify q e n t = q s e n = 0, q s t a t ≃ 1.935, and q r e l ≃ 1.4 . The situation corresponding to intermediate values of K, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG behavior and/or q -statistical behavior are observed.