ics, wave propagation, and seismic hazard analysis. Each of the latter summarizes the current understanding and articulates major goals and key questions for future research.
For present purposes, the term “dynamical system” can be understood to mean any set of coupled objects that obeys Newton’s laws of motion—rocks or tectonic plates, for example (1). If one can specify the positions and velocities of each of these objects at any given time and also know exactly what forces act on them, then the state of the system can be determined at a future time, at least in principle. With the advent of large computers, the numerical simulation of system behavior has become an effective method for predicting the behavior of many natural systems, especially in the Earth’s fluid envelopes (e.g., weather, ocean currents, and long-term climate change) (2). However, many difficulties face the application of dynamical systems theory to the analysis of earthquake behavior in the solid Earth. Forces must be represented as tensor-valued stresses (3), and the response of rocks to imposed stresses can be highly nonlinear. The dynamics of the continental lithosphere involves not only the sudden fault slips that cause earthquakes, but also the folding of sedimentary layers near the surface and the ductile motions of the hotter rocks in the lower crust and upper mantle. Moreover, because earthquake source regions are inaccessible and opaque, the state of the lithosphere at seismogenic depths simply cannot be observed by any direct means, despite the conceptual and technological breakthroughs described in Chapter 4.
From a geologic perspective, it is entirely plausible that earthquake behavior should be contingent on a myriad of mechanical details, most unobservable, that might arise in different tectonic environments. Yet earthquakes around the world share the common scaling relations, such as those noted by Gutenberg and Richter (Equation 2.5) and Omori (Equation 2.8). The intriguing similarities among the diverse regimes of active faulting make earthquake science an interesting testing ground for concepts emerging from the physics of complex dynamical systems. One consequence of recent interactions between these fields is that theoretical physicists have adopted a family of idealized models of earthquake faults as one of their favorite paradigms for a broad class of nonequilibrium phenomena (4). At the same time, earthquake scientists have become aware that earthquake faults may be intrinsically chaotic, geometrically fractal, and perhaps even self-organizing in some sense. As a result, an entirely new subdiscipline has emerged that is focused around the development and analysis of large-scale numerical simulations of deformation