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Abstract

Certain driven systems consisting of a large number of elements evolve towards a critical state with no characteristic time or length scales. This class of phenomena is described as Self-Organized Criticality (SOC). SOC relies on the condition of a slow driving of the systems and the existence of fast burst-like responses of them and includes earthquakes. We employ a model proposed by Xu et al. for ruptures in an elastic medium, subject to shear stress, and apply it to the study of earthquakes. In the model, the size of an earthquake is defined as the number of ruptures occurring sequentially on the basic units of discretization (squares) of the medium. A histogram of the earthquake sizes shows that the model is not completely scale invariant due to finite-size effects. To take them into account, we implement a finite-size-scaling analysis. The results of this analysis show that the model is scale invariant only when there is stress conservation. So, the model displays SOC in the conservative case only. We also study the dynamic quantities of the model, in particular the average stress in the system. The sets of average stress values are analyzed using two types of time series analysis. The nonlinear forecasting analysis investigates whether time series exhibit low-dimensional chaotic behavior as opposed to high-dimensional (or stochastic) behavior. We find that the above time series have a nonlinear structure, but with a substantial stochastic component, so SOC is inherently high-dimensional. The appearance of nonlinear structure is due to the fact that the system stops following linearly the external drive when it releases stress through earthquakes. The rescaled range analysis characterizes the time correlations (or memory effects) in the time series. We find strong positive time correlations in the above time series. Their presence is due to the nature of the driving in the model. These memory effects are destroyed as soon as a large earthquake resets the system.