Eur. Phys. J. Plus (2020) 135: 545
https://doi.org/10.1140/epjp/s13360-020-00558-7

Regular Article

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

¹ UR de Mécanique et de Modélisation des Systèmes Physiques (L2MSP), UFR/DSST, Université de Dschang, BP 69, Dschang, Cameroon
² UR Matière condensée, Electronique et Traitement du signal (UR-MACETS), Université de Dschang, BP 69, Dschang, Cameroon
³ Laboratory of Modelling and Simulation in Engineering and Biomimetics and Prototypes, University of Yaounde I, PO Box 812, Yaounde´, Cameroon
⁴ Department of Electrical and Electronic Engineering, Faculty of Engineering and Technology (FET), University of Buea, P.O. Box 63, Buea, Cameroon

^a tanekouguyb@yahoo.fr

Received: 28 July 2019
Accepted: 22 June 2020
Published online: 5 July 2020

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 Grünwald–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).