Structure and Dynamics of Polynomial Dynamical Systems

Details Structure and Dynamics of Polynomial Dynamical Systems Structure and Dynamics of Polynomial Dynamical Systems

2011-07-31

arxiv.org/abs/1108.0209v1

Biology

Quantitative Biology

Molecular Networks

10 pages, 3 figures. NSF CMMI Research and Innovation Conference 2011

Scientific paper

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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