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<Superscript>-α</Superscript>, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ<Subscript>M</Subscript> scales as λ<Subscript>M</Subscript> ∝ N<Superscript>-κ(α)</Superscript>, where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ<Subscript>M</Subscript>→0). In the short-range case, κ(α) 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 t<Subscript>c</Subscript> scales as t<Subscript>c</Subscript> ∝N<Superscript>β(α)</Superscript>, where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α> 1, and zero for α≥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 α-XY Hamiltonian ferromagnetic model. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2006