Simple hierarchical systems: Stability, self-organized criticality, and catastrophic behavior

A description of various kinds of behavior for a hierarchical model of defect development, representing a transition from stability to catastrophe, is suggested. It is shown that the self-organized criticality regarded as a linear form of the magnitude-frequency relation on a wide area of parameter corresponds to different kinds of system behavior. Examples of the systems representing self-organized criticality with stationary, periodic, and chaotic relation between the density of defects and the level of hierarchy are suggested. A complex behavior, when areas of self-organized criticality alternate with areas of catastrophe and/or stability, is observed in the model. Examples considered in the paper perform basic kinds of possible behavior on the transition interval from stability to catastrophe for a simple class of hierarchical systems.