From deterministic cellular automata to coupled map lattices

TL;DR

This paper introduces a universal mathematical framework to systematically construct coupled map lattices from cellular automata, revealing new bifurcation phenomena and unifying discrete and continuous dynamical systems.

Contribution

It presents a parameter-dependent method to derive coupled map lattices from cellular automata, generalizing CA rules and analyzing bifurcations in the resulting systems.

Findings

All deterministic CAs are encompassed in the RDCA framework as a special case.

A new bifurcation point is identified where deterministic CA dynamics become fuzzy.

Global homogeneous fixed points attract all initial conditions at large parameter values.

Abstract

A general mathematical method is presented for the systematic construction of coupled map lattices (CMLs) out of deterministic cellular automata (CAs). The entire CA rule space is addressed by means of a universal map for CAs that we have recently derived and that is not dependent on any freely adjustable parameters. The CMLs thus constructed are termed real-valued deterministic cellular automata (RDCA) and encompass all deterministic CAs in rule space in the asymptotic limit $\kappa \to 0$ of a continuous parameter $\kappa$. Thus, RDCAs generalize CAs in such a way that they constitute CMLs when $\kappa$ is finite and nonvanishing. In the limit $\kappa \to \infty$ all RDCAs are shown to exhibit a global homogeneous fixed-point that attracts all initial conditions. A new bifurcation is discovered for RDCAs and its location is exactly determined from the linear stability analysis of the global quiescent state. In this bifurcation, fuzziness gradually begins to intrude in a purely deterministic CA-like dynamics. The mathematical method presented allows to get insight in some highly nontrivial behavior found after the bifurcation.