Regular and chaotic motion in softened gravitational systems

The stability of the dynamical trajectories of softened spherical gravitational systems is examined, both in the case of the full $N$-body problem and that of trajectories moving in the gravitational field of non-interacting background particles. In the latter case, for $N \ge 10000$, some trajectories, even if unstable, had exceedingly long diffusion times, which correlated with the characteristic e-folding timescale of the instability. For trajectories of $N \approx 100000$ systems this timescale could be arbitrarily large --- and thus appear to correspond to regular orbits. For centrally concentrated systems, low angular momentum trajectories were found to be systematically more unstable. This phenomenon is analogous to the well known case of trajectories in generic centrally concentrated non-spherical smooth systems, where eccentric trajectories are found to be chaotic. The exponentiation times also correlate with the conservation of the angular momenta along the trajectories. For $N$ up to a few hundred, the instability timescales of $N$-body systems and their variation with particle number are similar to those of the most chaotic trajectories in inhomogeneous non-interacting systems. For larger $N$ (up to a few thousand) the values of the these timescales were found to saturate, increasing significantly more slowly with $N$. We attribute this to collective effects in the fully self-gravitating problem, which are apparent in the time-variations of the time dependent Liapunov exponents. The results presented here go some way towards resolving the long standing apparent paradoxes concerning the local instability of trajectories of gravitational systems (abridged).