On the Dynamics of Linear Finite Dynamical Systems Over Galois Rings

TL;DR

This paper extends the analysis of linear finite dynamical systems from cyclic modules to Galois rings and proposes algorithms to compute cycle lengths and transient structures.

Contribution

It introduces a novel extension of dynamical system analysis over Galois rings and provides algorithms for key dynamic properties.

Findings

Extended analysis to Galois rings.

Developed algorithms for cycle length computation.

Analyzed the structure of functional graphs.

Abstract

Linear finite dynamical systems play an important role, for example, in coding theory and simulations. Methods for analyzing such systems are often restricted to cases in which the system is defined over a field %and usually strive to achieve a complete description of the system and its dynamics. or lack practicability to effectively analyze the system's dynamical behavior. However, when analyzing and prototyping finite dynamical systems, it is often desirable to quickly obtain basic information such as the length of cycles and transients that appear in its dynamics, which is reflected in the structure of the connected components of the corresponding functional graphs. In this paper, we extend the analysis of the dynamics of linear finite dynamical systems that act over cyclic modules to Galois rings. Furthermore, we propose algorithms for computing the length of the cycles and the height of the trees that make up their functional graphs.