Granular convergence as an iterated local map

Granular convergence is a property of a granular pack as it is repeatedly sheared in a cyclic, quasistatic fashion, as the packing configuration changes via discrete events. Under suitable conditions the set of microscopic configurations encountered converges to a periodic sequence after sufficient shear cycles. Prior work modeled this evolution as the iteration of a pre-determined, random map from a set of discrete configurations into itself. Iterating such a map from a random starting point leads to similar periodic repetition. This work explores the effect of restricting the randomness of such maps in order to account for the local nature of the discrete events. The number of cycles needed for convergence shows similar statistical behavior to that of numerical granular experiments. The number of cycles in a repeating period behaves only qualitatively like these granular studies.