Periodicity, chaos and localization in a Burridge-Knopoff model of an earthquake with rate-and-state friction

We investigate the emergent dynamics when the slip law formulation of the non-linear rate-and-state friction law is attached to a Burridge-Knopoff spring-block model. We derive both the discrete equations and the continuum formulation governing the system in this framework. The discrete system (ODEs) exhibits both periodic and chaotic motion, where the system's transition to chaos is size-dependent, that is, how many blocks are considered. From the discrete model we derive the non-linear elastic wave equation by taking the continuum limit. This results in a non-linear partial differential equation (PDE) and we find that chaos ensues when the same parameter is increased. This critical parameter value needed for the onset of chaos in the continuous model is much smaller than the value needed in the case of a single block and we discuss the implications this has on dynamic modelling of earthquake rupture with this specific friction law. Most importantly, these results suggest that the friction law is scale-dependent, thus caution should be taken when attaching a friction law derived at laboratory scales to full-scale earthquake rupture models. Furthermore, we find solutions where the initial slip pulse propagates like a travelling wave, or remains localized in space, suggesting the presence of soliton and breather solutions. We discuss the significance of these pulse-like solutions and how they can be understood as a proxy for the propagation of the rupture front across the fault surface during an earthquake. We compute analytically the conditions for soliton solutions and by exploring the resulting parameter space, we introduce a possible method for determining a range of suitable parameter values to be used in future dynamic earthquake modelling.