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

Abstract

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 to (- alpha), being r the distance between maps. Thus, interactions are long-range (nonintegrable) when alpha is in [0,1], and short-range (integrable) when alpha is larger than 1. We verify that the largest Lyapunov exponent scales as lambda_M prop to N (- kappa(alpha)) ,where kappa(alpha) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit (hence lambda_M goes to 0). In the short-range case, kappa(alpha) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration tc scales as tc prop to N (- eta(alpha)), where eta(alpha) appears to be numerically in agreement with the following behavior: eta larger than 0 for alpha in [0,1), and zero for alpha larger than 1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the alpha-XY Hamiltonian ferromagnetic model.