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We revisit the r\^{o}le of discreteness and chaos in the dynamics ofself-gravitating systems by means of $N$-body simulations with active andfrozen potentials, starting from spherically symmetric stationary states andconsidering the orbits of single particles in a frozen $N$-body potential aswell as the orbits of the system in the full $6N$-dimensional phase space. Wealso consider the intermediate case where a test particle moves in the fieldgenerated by $N$ non-interacting particles, which in turn move in a staticsmooth potential. We investigate the dependence on $N$ and on the softeninglength of the largest Lyapunov exponent both of single particle orbits and ofthe full $N$-body system. For single orbits we also study the dependence on theangular momentum and on the energy. Our results confirm the expectation thatorbital properties of single orbits in finite-$N$ systems approach those oforbits in smooth potentials in the continuum limit $N \to \infty$ and that thelargest Lyapunov exponent of the full $N$-body system does decrease with $N$,for sufficiently large systems with finite softening length. However, singleorbits in frozen models and active self-consistent models have differentlargest 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.

$N$-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited