Evolution of Discrete Dynamical Systems

We investigate the evolution of three different types of discrete dynamical systems. In each case simple local rules are shown to yield interesting collective global behavior. (a) We introduce a mechanism for the evolution of growing small world networks. We demonstrate that purely local connection rules, when coupled with network growth, can result in short path lengths for the network as a whole. (b) We consider the general character of the spatial distributions of populations that grow through reproduction and subsequent local resettlement of new population members. Several simple one and two-dimensional point placement models are presented to illustrate possible generic behavior of these distributions. We show, both numerically and analytically, that all of the models lead to multifractal spatial distributions of population. (c) We present a discrete lattice model to investigate the segregation of three species granular mixtures in horizontally rotating cylinders. We demonstrate that the simple local rules of the model are able to reproduce many of the experimentally observed global phenomena