  We investigate the dependence of the largest Lyapunov exponent of a
$N$-particle self-gravitating ring model at equilibrium with respect to the
number of particles and its dependence on energy. This model has a continuous
phase-transition from a ferromagnetic to homogeneous phase, and we numerically
confirm with large scale simulations the existence of a critical exponent
associated to the largest Lyapunov exponent, although at variance with the
theoretical estimate. The existence of chaos in the magnetized state evidenced
by a positive Lyapunov exponent, even in the thermodynamic limit, is explained
by the resonant coupling of individual particle oscillations to the diffusive
motion of the center of mass of the system due to the thermal excitation of a
classical Goldstone mode. The transition from "weak" to "strong" chaos occurs
at the onset of the diffusive motion of the center of mass of the
non-homogeneous equilibrium state, as expected. We also discuss thoroughly for
the model the validity and limits of a geometrical approach for their
analytical estimate.
