Adebate has raged over whether fixed material and geometrical heterogeneities, or alternatively dynamic stress heterogeneities, arising through frictional instabilities dominate earthquake complexity. It may also be that both types of heterogeneities interact and are important. This paper makes a first step in examining this interaction, combining two previously separate lines of research. One line examined friction, which has attractors (the subset of the phase space that the system evolves towards in the long run) on homogeneous faults, which are simple, and then added fixed heterogeneities to the faults to obtain complex attractors. Another line examined frictions, which produced complex attractors on homogeneous faults. Here, we examine frictions, which produce complex attractors on homogeneous faults, and study them on heterogeneous faults, in order to study the interaction of dynamic stress heterogeneities and fixed fault heterogeneities. We consider two types of fixed heterogeneities: an additive noise and a multiplicative noise to the frictional strength of the fault. Because of the linearity of the bulk elastodynamics, the attractor is unaffected by additive fixed noise in the strength of the fault: adding an arbitrary function of space, fixed in time, to the friction leaves the resulting attractor unchanged. In contrast, multiplicative fixed noise multiplying the friction can have a profound effect on the resulting attractor. In the small multiplicative noise amplitude limit, the frictional weakening attractor is little perturbed; at finite amplitudes, fixed heterogeneities substantially alter the attractor. We see, as one consequence, a shift toward longer length events at larger amplitudes. Fixed heterogeneities are seen to reduce the irregularities created by the frictional instability we study, but by no means destroy them. We quantify this by examining a measure of variability of the importance in hazard estimates, the coefficient of variation of large event recurrence times. The coefficient of variation is seen to remain substantial even for large fixed heterogeneities. For friction that weakens with time, so the underlying uniform fault attractor is simple, fixed heterogeneities increase irregularity. For all frictions examined, at low fixed heterogeneity the stress concentrations left over by the ends of the large events dominate where most of the small events occur, while at higher heterogeneity the stress irregularities left over by fixed fault heterogeneities begin to dominate where the small events occur. This may be the strongest signature of fixed heterogeneities, and should be examined further in the Earth. Finally, in what may have important implications for more sophisticated estimates of earthquake hazard, we see a correlation of locations with lower strength drop having higher variation in large event repeat times.
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