Damage spreading and Lyapunov exponents in cellular automata

TL;DR

This paper investigates damage spreading and Lyapunov exponents in one-dimensional cellular automata using Boolean derivatives, revealing phase transitions akin to directed percolation, especially under noise.

Contribution

It introduces a Boolean derivative approach to quantify damage spreading and defines maximal Lyapunov exponents for cellular automata, linking chaos to phase transitions.

Findings

Chaotic rules follow a random matrix approximation.

A phase transition similar to directed percolation is observed.

Small noise induces the same transition in cellular automata.

Abstract

Using the concept of the Boolean derivative we study damage spreading for one dimensional elementary cellular automata and define their maximal Lyapunov exponent. A random matrix approximation describes quite well the behavior of ``chaotic'' rules and predicts a directed percolation-type phase transition. After the introduction of a small noise elementary cellular automata reveal the same type of transition.