On the Dynamics of Linear Finite Dynamical Systems Over Galois Rings

2026-04-01

AI Summary

This work addresses the limitations of existing methods in analyzing linear finite dynamical systems, which have largely been confined to fields and lack efficient algorithms for computing cycle lengths and transient heights. For the first time, the structural dynamics of such systems are extended to cyclic modules over Galois rings. By integrating Galois ring theory, decomposition of cyclic modules, and functional graph analysis, the authors develop an efficient algorithm that accurately computes the cycle length and tree height for each connected component of the state transition graph. This contribution not only generalizes the theoretical framework beyond fields but also substantially improves computational efficiency for key dynamical parameters, thereby providing a novel tool for studying dynamical systems over broader algebraic structures.

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.

Problem

• linear finite dynamical systems
• Galois rings
• functional graphs
• cycle length
• transient dynamics

Innovation

• linear finite dynamical systems
• Galois rings
• functional graphs
• cycle length
• transient height