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We investigate the dependence of the largest Lyapunov exponent of a$N$-particle self-gravitating ring model at equilibrium with respect to thenumber of particles and its dependence on energy. This model has a continuousphase-transition from a ferromagnetic to homogeneous phase, and we numericallyconfirm with large scale simulations the existence of a critical exponentassociated to the largest Lyapunov exponent, although at variance with thetheoretical estimate. The existence of chaos in the magnetized state evidencedby a positive Lyapunov exponent, even in the thermodynamic limit, is explainedby the resonant coupling of individual particle oscillations to the diffusivemotion of the center of mass of the system due to the thermal excitation of aclassical Goldstone mode. The transition from "weak" to "strong" chaos occursat the onset of the diffusive motion of the center of mass of thenon-homogeneous equilibrium state, as expected. We also discuss thoroughly forthe model the validity and limits of a geometrical approach for theiranalytical estimate.

Lyapunov Exponent and Criticality in the Hamiltonian Mean Field Model