Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of $N$ symplectically and globally coupled standard maps localized in a $d=1$ lattice array. The global coupling is modulated through a factor $r^{-\alpha }$, being $r$ the distance between maps. Thus, interactions are {\it long-range} (nonintegrable) when $0\le \alpha \le 1$, and {\it short-range} (integrable) when $\alpha >1$. We verify that the largest Lyapunov exponent ${\lambda }_M$ scales as $\lambda_{M} \propto N^{-\kappa(\alpha)}$,where$\kappa(\alpha)$ is positive when interactions are long-range, yielding {\it weak chaos} in the thermodynamic limit $N\to \mathrm{\infty }$ (hence ${\lambda }_M\to 0$). In the short-range case, $\kappa (\alpha )$ appears to vanish, and the behaviour corresponds to {\it strong chaos}. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration $t_c$ scales as $t_c\propto N^{\beta (\alpha )}$, where $\beta (\alpha )$ appears to be numerically consistent with the following behavior: $\beta >0$ for $0 \le \alpha < 1$,andzerofor$\alpha\ge 1$. All these results exhibit major conjectures formulated within nonextensive statistical mechanics (NSM). Moreover, they exhibit strong similarity between the present discrete-time system, and the $\alpha$-XY Hamiltonian ferromagnetic model, also studied in the frame of NSM.