ABSTRACT
A cellular automaton (CA) is a rule-based computing machine, which was first proposed by von Newmann in early 1950s and systematic studies were pioneered by Wolfram in 1980s. Since a cellular automaton consists of space and time, it is essentially equivalent to a dynamical system that is discrete in both space and time. The evolution of such a discrete system is governed by certain updating rules rather than differential equations. Although the updating rules can take many different forms, most common cellular automata use relatively simple rules (Von Newmann, 1966; Wolfram, 1983). On the other hand, an equation-based system such as the system of differential equations and partial differential equations also describe the temporal evolution in the domain. Usually, differential equations can also take different forms that describe various systems. Now one natural question is what is the relationship between a rule-based
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system and an equation-based system? Given differential equations, how can one construct a rule-based cellular automaton, or vice versa? There has been substantial amount of research in these areas in the past two decades. This chapter intends to summarize the results of the relationship among the cellular automata, partial differential equations (PDEs), and pattern formations.
