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 α model alone, and a strong stochasticity threshold (SST), found before for the β 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.