Slow-slip, slow earthquakes, period-two cycles, full and partial ruptures, and deterministic chaos in a single asperity fault

The assimilation of geological, geophysical, and laboratory data in physics-based models of fault dynamics promises increasingly realistic simulations of the seismic cycle. To assist this effort, I explore the dynamics of a single velocity-weakening asperity to delineate the relationship between a fault physical properties and the style of faulting during the seismic cycle based on a micro-physical model of rate-and-state friction. From a dimensional analysis, four non-dimensional parameters control the characteristics of slip events and the behavior during the interseismic period, including the Dieterich-Ruina-Rice number R-u, which is proportional to the ratio of asperity to critical nucleation size W/h*, and R-b = (b - a)/b, where a and b control the direct and local evolutionary effects, respectively. For sufficiently large R-b , the R-u number controls the spectrum of fault slip, from periodic slow-slip events to aperiodic fast ruptures. In finite faults, but not in semi-infinite faults, the transition between these end-members includes a bifurcation that involves period-two, period-four, period-six cycles, and deterministic chaos with increasing R-u Macroscopic fault slip is stabilized as R-b vanishes, even as R-u >> 1. For finite faults, but not for semi-infinite faults, the transition between slow and fast ruptures for increasing R-b includes another type of bifurcation with period-two and period-four cycles of slow and fast ruptures. For 0 < R-b << 1 and R-u >> 1, slow ruptures become chaotic, characterized by aperiodic bursts of slow earthquakes within a longer slow-slip episode. The third non-dimensional parameter is a cut-off velocity that affects the maximum slip speed. The static friction coefficient mu(0) strongly affects the rupture style, as fault strength provides an upper bound for stress drop. Many styles of instabilities and interseismic behaviors emerge depending on the coordinates in phase space, most importantly R-u and R-b. Combining complementary observations at different stages of the seismic cycle may offer an opportunity to infer these parameters in nature.