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 $0\leq\alpha\leq1$, 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 $N\to\infty$ (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 $0 \le \alpha < 1$, and zero for $\alpha\ge 1$. 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.