Slip instability development and earthquake nucleation as a dynamical system’s fixed-point attraction

Abstract:

A fault’s transition from slow creep to the propagation of an earthquake-generating dynamic rupture is thought to start as a quasi-static slip instability. Here we examine how such an instability develops on a sliding interface whose strength is governed by a slip rate- and state-dependent friction, where the state variable evolves according to the aging law.

We find that the development occurs as the attraction of a dynamical system to a fixed point. The fixed points are such that the state of slip and the rate at which velocity diverges (and its spatial distribution) are known. The fixed points are independent of the manner of external forcing and the values of slip rate and state before the onset of instability. For a fault under uniform normal stress and frictional properties, the sole parameter that determines the fixed point (to within a translational invariance) is the ratio of the frictional parameters, a/b (where, for steady-state rate weakening, 0<a/b<1). Below a critical value of a/b, the fixed points are asymptotically stable; however, stability is lost for a/b above that value. Increasing a/b above this critical value leads to a series of Hopf bifurcations. This cascade of bifurcations signals a quasi-periodic route to chaos, implying the existence of a second, larger, critical value of a/b (corresponding to the value at which the third Hopf bifurcation occurs), above which the slip instability may develop in a chaotic fashion. The fixed-point solutions, as well as the critical thresholds concerning their stability, depend on the configuration of slip (e.g., in/anti-plane or mixed-mode slip) and the elastic environment in which the interface is embedded (e.g., a slip surface between elastic half-spaces or one lying below and parallel to a free surface); solving for a fixed point reduces to the solution of an equivalent problem of an equilibrium slip-weakening fracture; and fixed-point stability is determined by linear stability analysis. Solutions of the fixed points and results concerning their stability are found either numerically or in closed form. For comparison, we find numerical solutions of instability development for given initial conditions and forcing. The resulting behaviors of the dynamical system conform precisely to expectations set by the fixed points and the analysis of their stability.