Fractional-order stability analysis of earthquake dynamics

Abstract

Fractional calculus is suitable for systems with memory and for fractal systems. Earthquakes have both properties. It is fair to claim that a fractional model is very efficient for earthquake modeling. Our study is focused on the effects of the fractional-order derivative on the 'train model' of Burridge–Knopoff. We note that these effects introduce additional degrees of freedom. Contrary to the integer model in which the fault remains seismologically active, the fractional derivative causes the transition from stick–slip oscillation to a stable equilibrium state. It is shown that the motion along the fault could be suppressed or reduced to an aseismic creeping when the fractional-order decreases. In the fractional model, we establish that the magnitude of the earthquake strongly depends on the fractional-order derivative. The stability of the stationary state is studied using fractional stability theory and the obtained results exhibit the powerful dependance of this stability on the fractional-order derivative. Chaos in the fractional-order system is controlled using a simple feedback controller. Furthermore, the finite-time stability of the fractional earthquake controller is investigated and sufficient conditions for the finite-time stability are presented. It appears that the estimated time of finite stability grows with the increment of the order of the fractional derivative. Moreover, the numerical simulations perfomed through the improved Adams–Bashforth–Moulton predictor–corrector scheme yield results that corroborate the theoretical predictions.