Criticality in a dynamic mixed system

We suggest a dynamic generalization of the simplest static hierarchical mixed model introduced by Shnirman and Blanter [Phys. Rev. Lett. 81, 5445 (1998); Phys. Rev. E. 60, 5111 (1998). We show that the stationary solution of the dynamic mixed model (DMM) demonstrates, in general, a linear form of the magnitude-frequency relation and may be considered a self-organized critical system. The dynamic mixed model demonstrates three principal kinds of system behavior: stability, catastrophe, and scale invariance. We show that the catastrophic area exists for all parameters of the mixture, and obtain three analytical expressions for boundary conditions of the stability and the scale invariance domains. As in the static model scale invariance appears as a result of a strong heterogeneity of the mixture. We describe how the magnitude-frequency relation reflects parameters of the heterogeneity and healing conditions for different domains of system behavior. Deviation of the DMM from the static mixed model and possible applications to earthquake prediction are discussed.