ON SELF-ORGANIZED CRITICALITY IN NONCONSERVING SYSTEMS

Two models with nonconserving dynamics and slow continuous deterministic driving, a stick-slip model (SSM) of earthquake dynamics and a toy forest-fire model (FFM), have recently been argued to show numerical evidence of self-organized criticality (generic, scale-invariant steady states). To determine whether the observed criticality is indeed generic, we study these models as a function of a parameter gamma which was implicitly tuned to a special value, gamma = 1, in their original definitions. In both cases, the maximum Lyapunov exponent vanishes at gamma = 1. We find that the FFM does not exhibit self-organized criticality for any gamma, including gamma = 1; nor does the SSM with periodic boundary conditions. Both models show evidence of macroscopic periodic oscillations in time for some range of gamma values. We suggest that such oscillations may provide a mechanism for the generation of scale-invariant structure in nonconserving systems, and, in particular, that they underlie the criticality previously observed in the SSM with open boundary conditions.