Abelian deterministic self-organized-criticality model: Complex dynamics of avalanche waves

Abstract

The aim of this study is to investigate a wave dynamics and a size scaling of avalanches which were created by the mathematical model [J. Černák, Phys. Rev. E 65, 046141 (2002)]. Numerical simulations were carried out on a two-dimensional lattice L×L in which two constant thresholds E(c)(I) = 4 and E(c)(II) > E(c)(I) were randomly distributed. The density of sites c of the thresholds E(c)(II) and threshold E(c)(II) are parameters of the model. Autocorrelations of avalanche size waves, Hurst exponents, avalanche structures, and avalanche size moments were determined for several densities c and thresholds E(c)(II). The results show correlated avalanche size waves and multifractal scaling of avalanche sizes not only for specific conditions, densities c = 0.0,1.0 and thresholds 8 ≤ E(c)(II) ≤ 32, in which relaxation rules were precisely balanced, but also for more general conditions, densities 0.0 < c < 1.0 and thresholds 8 ≤ E(c)(II) ≤ 32, in which relaxation rules were unbalanced. The results suggest that the hypothesis of a precise relaxation balance could be a specific case of a more general rule.