Phase space geometry and stochasticity thresholds in hamiltonian dynamics

Results of numerical computations of the largest Lyapunov exponent lambda(1)(varepsilon,N) as a function of the energy density varepsilon and the number of particles N are here reported for a Fermi-Pasta-Ulam alpha+beta model. These results show the coexistence at large N of two thresholds: a stochasticity threshold, found before for the alpha model alone, and a strong stochasticity threshold (SST), found before for the beta model alone. Although this coexistence may seem at first sight plausible, it is not obvious a priori that the alpha+beta model superimposes properties of the alpha and beta models independently. The main point of this paper, however, is a geometric characterization of the SST via the mean curvature of the constant energy hypersurfaces in the phase space of the model and the characteristic decay time of its time autocorrelation function tau(c)(varepsilon,N), which correlates with that of lambda(1)(varepsilon,N) for fixed N. This appears to provide important information on the very complicated geometry of the phase space of this simple solidlike model.