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Some statistical properties of finite-time stability exponents in thestandard map can be estimated analytically. The mean exponent averaged over theentire phase space behaves quite differently from all the other cumulants.Whereas the mean carries information about the strength of the interaction, andonly indirect information about dynamical correlations, the higher cumulantscarry information about dynamical correlations and essentially no informationabout the interaction strength. In particular, the variance and highercumulants of the exponent are very sensitive to dynamical correlations andeasily detect the presence of very small islands of regular motion via theiranomalous time-scalings. The average of the stability matrix' inverse trace iseven more sensitive to the presence of small islands and has a seeminglyfractal behavior in the standard map parameter. The usual accelerator modes andthe small islands created through double saddle node bifurcations, which comehalfway between the positions in interaction strength of the usual acceleratormodes, are clearly visible in the variance, whose time scaling is capable ofdetecting the presence of islands as small as 0.01% of the phase space. Westudy these quantities with a local approximation to the trace of the stabilitymatrix which significantly simplifies the numerical calculations as well asallows for generalization of these methods to higher dimensions. We alsodiscuss the nature of this local approximation in some detail.

Fluctuations of finite-time stability exponents in the standard map and the detection of small islands