Complex dynamics of elementary cellular automata emerging from chaotic rules

TL;DR

This paper introduces techniques to analyze and classify elementary cellular automata with chaotic rules by incorporating memory, showing that memory can transform chaotic systems into complex, predictable behaviors efficiently.

Contribution

It presents a novel approach to classify chaotic cellular automata based on their behavior with memory, enabling quick transformation from chaos to complexity without extensive computation.

Findings

Memory-enriched rules quickly converge to complex behaviors

Chaotic rules can be identified without exhaustive experiments

Analysis of glider dynamics reveals underlying structures

Abstract

We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behaviour. CA are well known computational substrates for studying emergent collective behaviour, 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 behaviour of any given function. Examples include mechanical computation, \lambda{} and Z-parameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behaviour 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 behaviour 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 analysis of well-known chaotic functions in one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.