  Some statistical properties of finite-time stability exponents in the
standard map can be estimated analytically. The mean exponent averaged over the
entire phase space behaves quite differently from all the other cumulants.
Whereas the mean carries information about the strength of the interaction, and
only indirect information about dynamical correlations, the higher cumulants
carry information about dynamical correlations and essentially no information
about the interaction strength. In particular, the variance and higher
cumulants of the exponent are very sensitive to dynamical correlations and
easily detect the presence of very small islands of regular motion via their
anomalous time-scalings. The average of the stability matrix' inverse trace is
even more sensitive to the presence of small islands and has a seemingly
fractal behavior in the standard map parameter. The usual accelerator modes and
the small islands created through double saddle node bifurcations, which come
halfway between the positions in interaction strength of the usual accelerator
modes, are clearly visible in the variance, whose time scaling is capable of
detecting the presence of islands as small as 0.01% of the phase space. We
study these quantities with a local approximation to the trace of the stability
matrix which significantly simplifies the numerical calculations as well as
allows for generalization of these methods to higher dimensions. We also
discuss the nature of this local approximation in some detail.
