On the Global Behavior of Some Systems of Difference

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 In Appendix .1, I give the unpublished results of the 21 systems that I studied.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 In Appendix .3, I give the unpublished results of the following system of rational difference equations: [Mathematical equations can not be displayed here, refer to PDF] where the parameters and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