  The parametric instability contribution to the largest Lyapunov exponent
(LLE) is derived for a mean-field Hamiltonian model, with attractive long-range
interactions. This uses a recent Riemannian approach to describe Hamiltonian
chaos with a large number N of degrees of freedom. Through microcanonical
estimates of suitable geometrical observables, the mean-field behavior of the
LLE is analytically computed and related to the second order phase transition
undergone by the system. It predicts that chaoticity drops to zero at the
critical temperature and remains vanishing above it, with the LLE scaling as
N^-1/3 to the leading order in N.
