Orbits in one-dimensional finite linear cellular automata

TL;DR

This paper analyzes the periodicity and relaxation behaviors of one-dimensional finite cellular automata with rules 90 and 150 using matrix representations and eigenvalue analysis to determine maximum periods and relaxation times.

Contribution

It introduces a matrix-based eigenvalue approach to characterize the maximum period and relaxation in finite cellular automata with specific rules.

Findings

Eigenvalue analysis determines maximum period lengths.

Matrix methods clarify relaxation times.

Results apply to rule-90 and rule-150 automata.

Abstract

Periodicity and relaxation are investigated for the trajectories of the states in one-dimensional finite cellular automata with rule-90 and 150. The time evolutions are described with matrices. Eigenvalue analysis is applied to clarify the maximum value of period and relaxation.