Logistic cellular automata

Abstract

Cellular Automata (CA), initially formalized to investigate self-reproducing constructions, are among the most frequently used tools to model and understand complex systems. These computational frameworks are de ned in discrete spacetime- state domains, where time evolution occurs through local interactions. Despite the simple properties and the succinct absence of long range connections, these implementations have been proven proper for studying large scale collective behavior and self-organizing mechanisms which often emerge in dynamical systems. Following the spirit of the well-known Logistic Map, we introduce a single parameter that tunes the dynamics of totalistic CA by mapping their discrete state space into a Cantor set. By introducing this simple approach on two archetypal models, this study addresses further investigation of several complex phenomena: critical deterministic phase transitions, pattern formation and tunable emanation of self-organized morphologies in these discrete domains. We rst apply this approach to Conway's Game of Life and observe sudden changes in asymptotic dynamics of the system accompanied by emergence of complex propagators. Incorporation of the new state space with system features is used to explain the critical points and formulate the tuning parameter range where the propagators adaptively survive, by investigating their autocatalytic local interactions. Similar behavior is present when the same recipe is applied to Rule 90, a totalistic elementary one-dimensional CA. In addition, the latter case shows that transitions between Wolfram's universality classes of CA can be achieved by tuning a single parameter continuously. Finally, we implement the same idea in other models and qualitatively report the expanding complexity that these frameworks support.