Phase space geometry and stochasticity thresholds in Hamiltonian dynamics

Results of numerical computations of the largest Lyapunov exponent lambda ( 1)(epsilon ,N) as a function of the energy density epsilon and the number o f particles N are here reported for a Fermi-Pasta-Ulam alpha+beta model. Th ese results show the coexistence at large N of two thresholds: a stochastic ity threshold, found before for the a model alone, and a strong stochastici ty threshold (SST), found before for the beta model alone. Although this co existence may seem at first sight plausible, it is not obvious a priori tha t the alpha+beta model superimposes properties of the a and beta models ind ependently. The main point of this paper, however, is a geometric character ization of the SST via the mean curvature of the constant energy hypersurfa ces in the phase space of the model and the characteristic decay time of it s time autocorrelation function tau (c)(epsilon ,N), which correlates with that of lambda (1)(epsilon ,N) for fixed iii. This appears to provide impor tant information on the very complicated geometry of the phase space of thi s simple solidlike model.