Dynamics of Linear Systems over Finite Commutative Rings

TL;DR

This paper extends the analysis of linear dynamical systems from finite fields to finite commutative rings, providing methods to determine cycle structures and improve understanding of system dynamics over these rings.

Contribution

It introduces a new method to analyze cycle structures of linear systems over finite commutative rings, building on previous algorithms for fixed point detection.

Findings

Developed a method to determine cycle structures of systems over finite rings.

Extended previous algorithms to analyze automorphisms over finite rings.

Enhanced understanding of the dynamics of linear systems beyond finite fields.

Abstract

The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous publication, the last two authors developed an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system or not. The algorithm can also be used to reduce the problem of finding the cycles of such a system to the case where the system is given by an automorphism. Here, we further analyze the cycle structure of such a system and develop a method to determine its cycles.