Abstract
We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behavior. CA are well-known computational substrates for studying emergent collective behavior, complexity, randomness and interaction between order and chaotic systems. A number of attempts have been made to classify CA functions on their space-time dynamics and to predict the behavior of any given function. Examples include mechanical computation, λ and Z-parameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behavior when they act in a historical mode, i.e. as CA with memory. We demonstrate that cell-state transition rules enriched with memory quickly transform a chaotic system converging to a complex global behavior from almost any initial condition. Thus, just in few steps we can select chaotic rules without exhaustive computational experiments or recurring to additional parameters. We provide an analysis of well-known chaotic functions in one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.
Highlights
In this paper we consider a simple tool to extract complex systems from a family of chaotic discrete dynamical system
We study elementary cellular automata (CA) (ECA) where each function evaluates a central cell with their two neighbourhoods and every cell takes a value of its binary alphabet
As well known as class of complex rules. In this classification class IV is of particular interest because the rules of the class exhibit non-trivial behaviour with rich diversity of patterns emerging and non-trivial interactions between travelling localizations, or gliders, e.g. ECA Rule 54 [Martınez et al, 2006]
Summary
In this paper we consider a simple tool to extract complex systems from a family of chaotic discrete dynamical system. We aim to base CA classification on their behaviour in a historical mode, i.e. as CA with memory [Alonso-Sanz, 2008]. A general classification of ECA was introduced in [Wolfram, 1994], as follows:. In this classification class IV is of particular interest because the rules of the class exhibit non-trivial behaviour with rich diversity of patterns emerging and non-trivial interactions between travelling localizations, or gliders, e.g. ECA Rule 54 [Martınez et al, 2006]. In present paper we aim to transform a chaotic evolution rule to a complex system by using memory memory chaotic ECA −−−−−−→ complex ECA and derive a new classes of CA functions with historic evolution. We believe that by employing historic evolution we are able to explore hidden properties of chaotic systems, and select chaotic rules with homogeneous dynamics
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