Eur. Phys. J. B 52, 493-500 (2006)
https://doi.org/10.1140/epjb/e2006-00327-2

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

¹ Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
² Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Ciudad Universitaria, CONICET, Córdoba, 5000, Argentina
³ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501, USA

Corresponding authors: ^a moyano@cbpf.br - ^b amajtey@famaf.unc.edu.ar - ^c tsallis@santafe.edu

Received: 22 February 2006
Published online: 21 August 2006

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^-α, 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_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→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_c scales as t_c ∝N^β(α), 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.