Date of Award

2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Gerasimos Ladas

Abstract

This dissertation is an exposition of systems of difference equations. I examine multiple examples of both piecewise and rational difference equations.

[Mathematical equations can not be displayed here, refer to PDF]

In the first two manuscripts, I share the published results of two members of the following family of 81 systems of piecewise linear difference equations:

where the initial condition (χ0, γ0) ∈R2, and where the parameters a b, c and d are integers between -1 and 1, inclusively. Since each parameter can be one of three values, there are 81 members. Each system is designated a number. The system’s number N is given by

N = 27(a + 1) + 9(b +1) + 3(c + 1) + (d +1) + 1.

The first manuscript is a study of System(2). System(2) results when a = b = c = -1 and d = 0. For System(2), I show that there exists a unique equilibrium solution and exactly two prime period-5 solutions, and that every solution of the system is eventually one of the two prime period-5 solutions or unique equilibrium solution.

The second manuscript is a study of System(8). System(8) results when a = b = -1, c = 1 and d = 0. For System(8), I show that there exists a unique equilibrium solution and exactly two prime period-3 solutions, and that except for the equilibrium solution, every solution of the system is eventually one of the two prime period-3 solutions.

Of the 81 systems, 65 have been studies thoroughly. In Appendix .1, I give the unpublished results of the 21 systems that I studied. In Appendix .2, I list all 81 systems (studied by W. Tikjha, E. Grove, G. Ladas, and E. Lapierre) each with a theorem or conjecture about its global behavior.

In the third manuscript, I give the published results of the following system of rational difference equations:

[Mathematical equations can not be displayed here, refer to PDF]

where the parameters and initial conditions are positive real values. I show that the system is permanent and has a unique positive equilibrium which is locally asymptotically stable. I also find sufficient conditions to insure that the unique positive equilibrium is globally asymptotically stable.

In Appendix .3, I give the unpublished results of the following system of rational difference equations:

[Mathematical equations can not be displayed here, refer to PDF]

where the parameters and initial conditions are positive real values. I show that the system is permanent. I also find sufficient conditions to insure that the unique positive equilibrium is globally asymptotically stable.

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