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

Abstract

The aim of this study is to investigate a wave dynamics and size scaling of avalanches which were created by the mathematical model {[}J. Černák Phys. Rev. E \textbf{65}, 046141 (2002)]. Numerical simulations were carried out on a two dimensional lattice $L\times L$ in which two constant thresholds $E_c^I=4$ and $E_c^{II}>E_c^I$ were randomly distributed. A density of sites $c$ with the threshold $E_c^{II}$ and threshold $E_c^{II}$ are parameters of the model. I have determined autocorrelations of avalanche size waves, Hurst exponents, avalanche structures and avalanche size moments for several densities $c$ and thresholds $E_c^{II}$. I found correlated avalanche size waves and multifractal scaling of avalanche sizes not only for specific conditions, densities $c=0.0$, 1.0 and thresholds $8\le E_c^{II}\le 32$, in which relaxation rules were precisely balanced, but also for more general conditions, densities $0.0<c<1.0$ and thresholds $8\leq E_{c}^{II}\leq3 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.

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