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.