Complex dynamics in the two spring-block model for earthquakes with fractional viscous damping

Abstract

Fractional calculus is suitable to model complex systems with memory and fractal systems. Earthquakes are both complex systems with long-memory and some of their faults have fractal properties. It is fair to claim that a fractional model is very efficient for earthquake modeling. The stability analysis is performed to determine the domain in which the system is stable or not. The model gives us an excellent interpretation of the earthquake as a stick–slip motion. The numerical simulation method used is that of Grunwald–Letnikov which is based on the generalization of the classical derivative. Nonlinear dynamical tools such as bifurcation diagrams and 0–1 tests are used to analyze the dynamics of the considered two spring-block system as an analogy of a fault with two interacting patches or asperities. Our results show that fractional-order derivative and viscosity have a significant influence on the recurrence time of an earthquake. The presence of fluid between the blocks and stickiness surface increases the recurrence of an earthquake. High viscosity leads to a slow slip or silent earthquakes. We observe that the introduction of the fractional derivative can extend or delay the transition from stick–slip (seismic) motion to an equilibrium state (aseismic fault).