# Stabilization Bounds for Linear Finite Dynamical Systems

## TL;DR

This paper investigates the long-term behavior of linear finite dynamical systems by establishing upper bounds on the number of iterations needed for systems to reach fixed points, aiding in analysis without exhaustive enumeration.

## Contribution

The paper introduces two novel upper bounds on iteration counts for linear finite dynamical systems, applicable to fixed point detection and based on submodule and modular properties.

## Key findings

- Derived two upper bounds for system stabilization time
- Bounds are applicable to fixed point system testing
- Examples demonstrate bounds' optimality

## Abstract

A common problem to all applications of linear finite dynamical systems is analyzing the dynamics without enumerating every possible state transition. Of particular interest is the long term dynamical behaviour. In this paper, we study the number of iterations needed for a system to settle on a fixed set of elements. As our main result, we present two upper bounds on iterations needed, and each one may be readily applied to a fixed point system test. The bounds are based on submodule properties of iterated images and reduced systems modulo a prime. We also provide examples where our bounds are optimal.