Abstract:
Recently, Rice [1993] pointed out that, up to now, the self-organizing models which have produced complex nonperiodic sequences of events have all been sensitive to the spatial discretization used, and thus did not have a well defined continuum limit. He went on the suggest that spatial nonuniformity or ''inherent discreteness'' may be a necessary ingredient in allowing the complexity to develop in these systems. In this paper, I present a counterexample to this suggestion: a spatially uniform model with a well defined continuum limit is shown to give rise to complex nonperiodic sequences. The complexity arises in the deterministic model from inertial dynamics with a velocity-weakening frictional instability, with the instability being stabilized at short lengthscales by a viscous term. The numerical results are shown to be independent of the spatial discretization for discretizations small compared to the viscous lengthscale. Furthermore, the qualitative features of the complexity produced are seen to be invariant with respect to two very different types of small scale cutoffs, implying a universality of the results with respect to the details of the small scale cutoff.