Real Linear Automata with a Continuum of Periodic Solutions for Every Period

Abstract

Interesting dynamical features such as periodic solutions of binary cellular automata are rare and therefore difficult to find, in general. In this paper, we illustrate an effective method in identifying fixed and periodic points of traditional one- and two-dimensional [Formula: see text]-valued cellular automaton systems, using cycle graphs. We also show that when the binary or [Formula: see text]-valued cellular states are extended to real-values and when defined as doubly-infinite vectors, there exists a continuum of periodic solutions for every period, even when the governing local rules are simply linear. In addition, we demonstrate the important effect that boundary conditions have on systems’ dynamical structures.