This book provides an overview of the main approaches used to analyze
the dynamics of cellular automata. Cellular automata are an
indispensable tool in mathematical modeling. In contrast to classical
modeling approaches like partial differential equations, cellular
automata are relatively easy to simulate but difficult to analyze. In
this book we present a review of approaches and theories that allow the
reader to understand the behavior of cellular automata beyond
simulations. The first part consists of an introduction to cellular
automata on Cayley graphs, and their characterization via the
fundamental Cutis-Hedlund-Lyndon theorems in the context of various
topological concepts (Cantor, Besicovitch and Weyl topology). The second
part focuses on classification results: What classification follows from
topological concepts (Hurley classification), Lyapunov stability (Gilman
classification), and the theory of formal languages and grammars (Kůrka
classification)? These classifications suggest that cellular automata be
clustered, similar to the classification of partial differential
equations into hyperbolic, parabolic and elliptic equations. This part
of the book culminates in the question of whether the properties of
cellular automata are decidable. Surjectivity and injectivity are
examined, and the seminal Garden of Eden theorems are discussed. In
turn, the third part focuses on the analysis of cellular automata that
inherit distinct properties, often based on mathematical modeling of
biological, physical or chemical systems. Linearity is a concept that
allows us to define self-similar limit sets. Models for particle motion
show how to bridge the gap between cellular automata and partial
differential equations (HPP model and ultradiscrete limit). Pattern
formation is related to linear cellular automata, to the Bar-Yam model
for the Turing pattern, and Greenberg-Hastings automata for excitable
media. In addition, models for sand piles, the dynamics of infectious d
