Structure and Dynamics of Polynomial Dynamical Systems

Source: arXiv

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.

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