and Manufacturing Innovation STRUCTURE AND DYNAMICS OF POLYNOMIAL DYNAMICAL SYSTEMS ∗

Discrete models have a long tradition in engi-neering, 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 depen-dency graph, that is, the directed graph that indicates the variable dependencies, provides a mapping from systems to graphs. A basic prop-erty of this mapping is derived and used to prove that dynamical systems with an acyclic depen-dency 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. ∗This work was supported by the National Science Foundation under Grant Nr. CMMI-0908201