Dynamical origin of spatial order

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Abstract
We present a numerical study of a one-dimensional version of the Burridge-Knopoff model [Bull. Seismol. Soc. Am. 57, 341 (1967)] with stick-slip dynamics. The solutions of the model in the low velocity regime represent earthquakes in a simple transform fault and have chaotic behavior [J. M. Carlson and J. S. Langer, Phys. Rev. Lett. 62, 2632 (1989); Phys. Rev. A 40, 6470 (1989)]. It has been shown recently that in a higher velocity regime there are solutions of the model with periodic boundary conditions that are solitonlike and not necessarily chaotic [J. Schmittbuhl, J. P. Vilotte, and S. Roux, Europhys. Lett. 21, 374 (1993)]. We show here that stable, nearly periodic solutions also exist in a certain window of parameter space when the model has free boundary conditions. These solutions are periodic in both time and space and display striation effects that are strikingly similar to those seen experimentally by Gollub and co-workers [Phys. Rev. A 43, 811 (1991); Phys. Rev. E 47, 820 (1993)]. For an arbitrary disordered set of initial conditions, the short-time behavior is noisy, but the stable nearly periodic solutions emerge in the long-time limit. We discuss the origin of the window and show that the nature of the solution found depends strongly on the boundary condition. We also discuss the effects of symmetry breaking and disorder and show that even in a highly disordered regime the system can spontaneously organize itself so that very nearly stable noise-free solutions emerge.