Statistical characterization of the standard map

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(ent), the sensitivity to initial conditions q(sen), the distribution of a time-averaged (over successive iterations) phase-space coordinate q(stat), and the relaxation to the equilibrium final state q(rel), collapse onto a fixed point, i.e. q(ent) = q(sen) = q(stat) = q(rel) = 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(ent) = q(sen) = 0, qstat similar or equal to 1.935, and q(rel) similar or equal to 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.