Discreteness effects, $N-$body chaos and the onset of radial-orbit
  instability
  We study the stability of a family of spherical equilibrium models of
self-gravitating systems, the so-called $\gamma-$models with Osipkov-Merritt
velocity anisotropy, by means of $N-$body simulations. In particular, we
analyze the effect of self-consistent $N-$body chaos on the onset of
radial-orbit instability (ROI). We find that degree of chaoticity of the system
associated to its largest Lyapunov exponent $\Lambda_{\rm max}$ has no
appreciable relation with the stability of the model for fixed density profile
and different values of radial velocity anisotropy. However, by studying the
distribution of the Lyapunov exponents $\lambda_{\rm m}$ of the individual
particles in the single-particle phase space, we find that more anisotropic
systems have a larger fraction of orbits with larger $\lambda_{\rm m}$.
