Self-Organized Criticality With Complex Scaling Exponents In The Train Model

The train model, which is a variant of the Burridge-Knopoff earthquake model, is investigated for a velocity-strengthening friction law. It shows self-organized criticality with complex scaling exponents. That is, the probability density function of the avalanche strength is a power law times a log-periodic function. Exact results (scaling exponent: 3/2 + 2 pi i/ln 4) are found for a nonlocal cellular automaton that approximates the overdamped train model. Further the influence of random static friction is discussed.