<p>In models of faults as elastic continua with a frictional interface, earthquake nucleation is the initiation of a&#160;propagating dynamic fault rupture nucleated by a localized slip instability. A mechanism capturing both&#160;the weakening process leading to nucleation as well as fault healing between events, is a slip rate- and state-dependent friction, with so-called direct effect and evolution effects [Dieterich, JGR 1979; Ruina,&#160;JGR 1983]. While the constitutive representation of the direct effect is theoretically supported [e.g.,&#160;Nakatani, JGR 2001; Rice et al., JMPS 2001], that of the evolution effect remains empirical and a number&#160;of state-evolution laws have been proposed to fit lab rock friction data [Ruina, JGR 1983; Kato and Tullis,&#160;GRL 2001; Bar-Sinai et al., GRL 2012; Nagata et al., JGR 2012]. These laws may share a common&#160;linearization about steady-state, such that a linear stability analysis of steady, uniform sliding yields a&#160;single critical wavelength for unstable growth of perturbations [Rice and Ruina, JAM 1983]. However, the&#160;laws&#8217; differences are apparent at later, non-linear stages of instability development.</p><div>Previously, we showed that instability development under aging-law state evolution could be understood&#160;in terms of dynamical systems [Viesca, PR-E 2016, PRS-A 2016]: the non-linear acceleration of slip&#160;occurs as the attraction of a fault&#8217;s slip rate to a fixed point, corresponding to slip rate diverging with a&#160;fixed spatial distribution and rate of acceleration. Here we show that this framework can also be applied&#160;to understand slip instability development under all commonly used evolution laws, including the so-called&#160;slip and Nagata laws. To do so, we develop an intermediate state evolution law that transitions&#160;between the slip and aging laws with the adjustment of a single parameter. We show that, to within a&#160;variable transformation, the intermediate law is equivalent to the Nagata law and that fixed-point blow-up&#160;solutions exist for any value of the transition parameter. We assess these fixed-points&#8217; stability via a linear&#160;stability analysis and provide an explanation for previously observed behavior in numerical solutions for&#160;slip rate and state evolution under various evolution laws [Ampuero and Rubin, JGR 2008; Kame et al.,&#160;2013; Bar-Sinai et al., PR-E 2013; Bhattacharya and Rubin, JGR 2014].</div>
Shrinking targets in parametrised families