GENERALIZED ONE-DIMENSIONAL CELLULAR AUTOMATA AND DISCRETE DYNAMIC SYSTEMS

Abstract

Cellular automata (CA) are a research subject of plethora of works. CA are applied in many areas of natural science and suitable for modeling of difference processes, phenomenon (Belousov-Zhabotinskiy’s Reaction [2], the turbulence [3], epidemic spread [4], population dynamic [5], electoral pro-cesses [6], etc.). The 30th rule of Wolfram’s Code [1] is widely spread in computer sciences for creating of pseudorandom sequences of integers. As known, the CA is a dynamic system. Usually CA set by logical op-erations, set of rules. It is quite cumbersome for program realization and impact on debugging time. The easier way is creation of the algorithm by analytic representation and don’t use enormous amount of loops. Object of this work is representation of CA as a system of difference equations. It allows us to generalize CA, thus we could model it in spaces of any dimensions and it cells could take an arbitrary finite or infinite num-ber of states. Further we would apply famous control methods of dynamical systems using the information ofprevious states of the system. Besides, this approach helps us to consider CA wider and find new features of these systems, regular structures, periodic structures or structures that are close to periodic ones. In this work we use the superposition of linear and nonlinear maps that calls as Diffusion and Reaction according-ly. This approach is applied on CA that work as Wolfram Code [1] respectively. Examples of suggested ap-proach applying are illustrated in the last section. All in all we have represented CA in form of a system of difference equations. In further study it would allow to research phenomenon connected with CA construc-tion or we could control CA due to method represented in [7, 8].