Discrete system

Abstract: A discrete event system (DES) is a dynamic system that evolves in accordance with the abrupt occurrence, at possibly unknown irregular intervals, of physical events. Such systems arise in a variety of contexts ranging from computer operating systems to the control of complex multimode processes. A control theory for the logical aspects of such DESs is surveyed. The focus is on the qualitative aspects of control, but computation and the related issue of computational complexity are also considered. Automata and formal language models for DESs are surveyed. >

Abstract: Three-dimensional analyses are reported of slip on a long vertical strike-slip fault between steadily driven elastic crustal blocks. A rate- and state-dependent friction law governs motion on the fault; the law includes a characteristic slip distance L for evolution of surface state and slip weakening. Because temperature and normal stress vary with depth, frictional constitutive properties (velocity weakening/ strengthening) do also. Those properties are taken either as uniform along-strike at every depth or as perturbed modestly from uniformity. The governing equations of quasi-static elasticity and frictional slip are solved on a computational grid of cells as a discrete numerical system, and a viscous radiation damping term is included to approximately represent inertial control of slip rates during earthquakelike instabilities. The numerical results show richly complex slip, with a spectrum of event sizes, when solved for a grid with oversized cells, that is, with cell size h that is too large to validly represent the underlying continuous system of equations. However, in every case for which it has been feasible to do the computations (moderately large L only), that spatio-temporally complex slip disappears in favor of simple limit cycles of periodically repeated large earthquakes with reduction of cell size h. Further study will be necessary to determine whether a similar trhnsition occurs when the elastodynamics of rupture propagation is treated more exactly, rather than in the radiation damping approximation. The transition from complex to ordered fault response occurs as h is reduced below a theoretically derived nucleation size h* which scales with L but is 2 x 104 to 105 larger in cases considered. Cells larger than h* can fail independently of one another, whereas those much smaller than h* cannot slip unstably alone and can do so only as part of a cooperating group of cells. The results contradict an emergent view that spatio-temporal complexity is a generic feature of mechanical fault models. Such generic complexity does apparently result from models which are inherently discrete in the sense of having no well-defined continuum limit as h diminishes. Those models form a different class of dynamical systems from models like the present one that do have a continuum limit. Strongly oversized cells cause the model developed here to mimic an inherently discrete system. It is suggested that oversized cells, capable of failing independently of one another, may crudely represent geometrically disordered fault zones, with quasi-independent fault segments that join one another at kinks or jogs. Such geometric disorder, at scales larger than h*, may force a system with a well-defined continuum limit to mimic an inherently discrete system and show spatio-temporally complex slip at those larger scales.