Structure and Dynamics of Polynomial Dynamical Systems

TL;DR

This paper introduces an algebraic framework for analyzing polynomial dynamical systems over finite fields, revealing how the structure of the dependency graph constrains the system's dynamics, such as fixed points and periodic orbits.

Contribution

It establishes a connection between the dependency graph structure and the dynamical behavior of polynomial systems, providing new theoretical insights and applications.

Findings

Acyclic dependency graphs imply a unique fixed point.

Systems with cyclic graphs can have multiple fixed points or periodic orbits.

Application to virus competition model demonstrates practical relevance.

Abstract

Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This paper discusses an algebraic framework to study such questions. The systems discussed here are given by mappings on an affine space over a finite field, whose coordinate functions are polynomials. They form a general class of models which can represent many discrete model types. Assigning to such a system its dependency graph, that is, the directed graph that indicates the variable dependencies, provides a mapping from systems to graphs. A basic property of this mapping is derived and used to prove that dynamical systems with an acyclic dependency graph can only have a unique fixed point in their phase space and no periodic orbits. This result is then applied to a published model of in vitro virus competition.