$N$-body chaos and the continuum limit in numerical simulations of
  self-gravitating systems, revisited
  We revisit the r\^{o}le of discreteness and chaos in the dynamics of
self-gravitating systems by means of $N$-body simulations with active and
frozen potentials, starting from spherically symmetric stationary states and
considering the orbits of single particles in a frozen $N$-body potential as
well as the orbits of the system in the full $6N$-dimensional phase space. We
also consider the intermediate case where a test particle moves in the field
generated by $N$ non-interacting particles, which in turn move in a static
smooth potential. We investigate the dependence on $N$ and on the softening
length of the largest Lyapunov exponent both of single particle orbits and of
the full $N$-body system. For single orbits we also study the dependence on the
angular momentum and on the energy. Our results confirm the expectation that
orbital properties of single orbits in finite-$N$ systems approach those of
orbits in smooth potentials in the continuum limit $N \to \infty$ and that the
largest Lyapunov exponent of the full $N$-body system does decrease with $N$,
for sufficiently large systems with finite softening length. However, single
orbits in frozen models and active self-consistent models have different
largest Lyapunov exponents and the $N$-dependence of the values in non-trivial,
so that the use of frozen $N$-body potentials to gain information on large-$N$
systems or on the continuum limit may be misleading in certain cases.
