ON SELF-ORGANIZED CRITICALITY IN NONCONSERVING SYSTEMS

Two models with nonconserving dynamics and slow continuous determinist ic driving, a stick-slip model (SSM) of earthquake dynamics and a toy forest-fire model (FFM), have recently been argued to show numerical e vidence of self-organized criticality (generic, scale-invariant steady states). To determine whether the observed criticality is indeed gene ric, we study these models as a function of a parameter gamma which wa s implicitly tuned to a special value, gamma = 1, in their original de finitions. In both cases, the maximum Lyapunov exponent vanishes at ga mma = 1. We find that the FFM does not exhibit self-organized critical ity for any gamma, including gamma = 1; nor does the SSM with periodic boundary conditions. Both models show evidence of macroscopic periodi c 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, tha t they underlie the criticality previously observed in the SSM with op en boundary conditions.