  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 $S_1=S_{BG} \equiv
-k\sum_{i=1}^W p_i \ln p_i$) grows linearly with time before achieving its
$W$-dependent saturation value; $q_{\text{sen}}$ 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 $e_q^z \equiv[1+(1-q)z]^{1/(1-q)}$;
$q_{\text{stat}}$ is the index associated with the $q_{\text{stat}}$-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
$q_{\text{rel}} \simeq1.4$. 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.
