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\title{Chaotic asteroidal trajectories \\
exhibiting multiple bursts of eccentricity: \\
a statistical analysis}
\author{Ivan~I.~Shevchenko \\
Institute of Theoretical Astronomy, \\
Russian Academy of Sciences, St.~Petersburg, Russia \\ \\
Hans~Scholl \\
Observatoire de la C\^ote d'Azur, \\ Department G.~D.~Cassini,
Nice, France}

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\begin{document}
\maketitle

The phenomenon of  eccentricity jumps discovered by Wisdom~(1982)
in the long-term behaviour of asteroids in the 3/1 Kirkwood gap is
considered in the planar-elliptic Sun--Jupiter--asteroid problem.
A statistical analysis of such intermittent trajectories
in the 3/1 gap, displaying numerous up and down transitions
between low and high eccentricity modes, is
performed. The distribution of time intervals $N$
between jumps, measured in units of an asteroid's revolution of
perihelion, is derived for several intermittent trajectories.
For smaller values of $N$ (satisfying an approximate limit of
$N < 50$), the distribution is found to be of Poisson type,
while in its tail, the distribution can be described
by a power law $N^{p}$. The index $p$ lies in the interval
$-1.9 \leq p \leq -1.5$ for the computed trajectories.
The power-law character of the distribution and numerical
values of the power-law index are in accordance with
numerical experiments by Chirikov and Shepelyansky~(1981, 1986),
and are in accordance with theoretical results
obtained by Chirikov~(1990) and Murray~(1991)
on statistics of Poincar\'e recurrences for the separatrix
mapping, which describes the motion in the vicinity of the
separatrix of a nonlinear resonance. The similarity between the
distributions for the separatrix mapping and for asteroid motion
in the 3/1 gap suggests
that the separatrix mapping controls most general properties
of intermittent asteroidal motion. The Poisson
character which we found in the ``main'' part of
the distribution is
discussed and explained.

\end{document}
