Gary L. Beck

Date of Award

12-1-2006

Document Type

Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

First Advisor

Dr. Dora Matache

Abstract

This study considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number of k parent nodes. The network is analyzed when the connectivity k is fixed or variable. This is an extension of a model studied by Andrecut [4) who considered the dynamics of two stochastic coupled maps. Making use of the Boolean rule that is a generalization of Rule 22 of elementary cellular automata, a generalization of the formula for providing the probability of finding a node in state 1 at a time t is determined and used to generate consecutive states of the network for both the real system and the model. We show that the model is a good match for the real system for some parameter combinations. However, our results indicate that in some cases the model may not be a good fit, thus contradicting results of [4]. For valid parameter combinations, we also study the dynamics of the network through Lyapunov exponents, bifurcation diagrams, fixed point analysis and delay plots. We conclude that for fixed connectivity the model is a more reliable match for the real system than in the case of variable connectivity, and that the system may exhibit stability or chaos depending on the underlying parameters. In general high connectivity is associated with a convergence to zero of the probability of finding a node in state 1 at time t.

Comments

A Thesis Presented to the Department of Mathematics and the Faculty of the Graduate College University of Nebraska In Partial Fulfillment of the Requirements for the Degree Master of Arts University of Nebraska at Omaha. Copyright 2006 Gary L. Beck

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