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strongly pinned and weakly pinned regions in U(x, t)—is a generic feature of the models we study. However, in our case, this irregularity is an intrinsic property of the dynamics, as opposed to some extrinsic property introduced by the modeler.

In this regard, our point of view is the opposite of that taken by Ben-Zion and Rice (6, 12, 23), who see no evidence in their calculations that slip complexity can be generated solely by nonlinear dynamics on smooth faults. They are able, however, to reproduce observed behaviors by making their models inherently discrete and/or heterogeneous. There are some fundamental differences between their calculations and ours. Ben-Zion and Rice use rate- and state-dependent friction laws that do not produce the strong instabilities at high slipping speeds that emerge from our velocity- or slip-weakening laws. Also, in the work that they have published to date, they use quasi-static approximations, whereas we must solve the equations of motion for fully inertial elastodynamics in order to see the instabilities that generate complexity.

We cannot claim to know which of these approaches ultimately will prove to be the more realistic and useful, but we do believe that the differences in friction and dynamics explain the discrepancies between our results and those of others. If the geometric complexity of real faults is the overwhelmingly most important source of slip complexity, then quasistatic models with externally imposed heterogeneities may be most useful for practical purposes. On the other hand, if intrinsically smooth faults are deterministically chaotic systems that generate their own irregularities during unstable slipping motions, then models of the sort that we have described here will be essential for progress in earthquake prediction.

The work of J.M.C. was supported by the David and Lucile Packard Foundation, National Science Foundation (NSF) Grant DMR-9212396, and a grant from Los Alamos National Laboratory. J.S.L. was supported by Department of Energy Grant DE-FG03–84ER45108 and NSF Grant PHY89–04035. C.R.M. was supported by NSF Grant ASC-9309833 and the Cornell Theory Center. B.E.S. was supported by NSF Grants 93–16513 and USC-569934.

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