Statistical characterization of the standard map

The standard map, paradigmatic conservative system in the $(x,p)$ phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter $K$. A detailed numerical description is achieved in the present paper. More precisely, for large values of $K$, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=q_{\text{stat}}=q_{\text{rel}}=1$, where $q_{\text{ent}}$ is the $q$-index for which the nonadditive entropy $S_q \equiv k \frac{1-\sum_{i=1}^W p_i^q}{q-1}$(with$S_1=S_{BG} \equiv -k\sum_{i=1}^W p_i \ln p_i$)growslinearlywithtimebeforeachievingits$W$-dependentsaturationvalue;$q_{$\text{sen}$}$ characterizes the time increase of the sensitivity $\xi$ to the initial conditions, i.e., $\xi \sim e_{q_{$\text{sen}$}}^{\lambda_{q_{$\text{sen}$}} ,t};(\lambda_{q_{$\text{sen}$}}>0)$,where$e_q^z \equiv[1+(1-q)z]^{1/(1-q)}$;$q_{$\text{stat}$}$istheindexassociatedwiththe$q_{$\text{stat}$}$-Gaussian distribution of the time average of successive iterations of the $x$-coordinate; finally, $q_{\text{rel}}$ characterizes the $q_{\text{rel}}$-exponential relaxation with time of the entropy $S_{q_{\text{ent}}}$ towards its saturation value. In remarkable contrast, for small values of $K$, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with $q$-statistics, we verify $q_{\text{ent}}=q_{\text{sen}}=0$, $q_{\text{stat}}\simeq 1.935$, and $q_{\text{rel}}\simeq 1.4$. 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 or $q$-statistical behavior are observed.