Date of Award

2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Mustafa R. S. Kulenović

Abstract

The focus of this thesis is on some contemporary problems in the field of difference equations and discrete dynamical systems. The problems that I worked on range global attractivity results to all types of bifurcations for systems of difference equations in the plane.

The major goal was to investigate the impact of nonlinear perturbation and the introduction of quadratic terms on linear fractional difference equations such as the Beverton-Holt as well as the Sigmoid Beverton-Holt Model with delay that describes the growth or decay of single species.

The first Manuscript was on the study of the following equation:

xn+1 = p + x2n / x2n-1 (1)

Which was an open problem suggested by Dr. Kulenović. It is a a perturbation of the linear fractional difference equation:

xn+1 = p + x2n / x2n-1

The solutions of Equation(1) undergo very interesting dynamics as I showed that the variation of the parameter p can cause the exhibition of the Naimark-Sacker bifurcation. I compute the direction of the Naimark-Sacker bifurcation for the di erence equation (1) and I provide an asymptotic approximation of the closed invariant curve which comes to existence as the unique positive equilibrium point loses its stability. Moreover tools and global stability result to provide a region of the parameter where local stability implies global stability of the equilibrium.

In my second Manuscript, I considered the difference equation:

xn+1 = xn / Cx2n-1 + Dxn + F (2)

where C;D and F are positive numbers and the initial conditions x-1 and x0 are non-negative numbers. Equation (2) which is also a non-linear perturbation of the Beverton Holt model, belongs to the category of difference equations with a unique positive equilibrium that exhibit the Naimark-Sacker bifurcation. The investigation of the dynamics of such equation is very challenging as it depends on more than one parameter. However I give a method for proving that its dynamics undergoes the Naimark-Sacker bifurcation. Moreover I compute the direction of the Neimark-Sacker bifurcation for this difference equation and provide the asymptotic approximation of the invariant closed curve. Furthermore I give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.

The following theorem is the major result that I relied on to prove global asymptotic stability of the equilibria in my first two Manuscripts:

Theorem 1 Let I be a compact interval of the real numbers and assume that f : I3 --> I is a continuous function satisfying the following properties:

  1. f(x, y, z) is non-decreasing in x and non-increasing in y and z
  2. The system [f(M, m, m) = M f(m, M, M) = m has a unique M = m in I.

Then the equation xn+1 = f(xn, xn-1, xn-2) has a unique equilibrium x in I and every solution of it that enters I must converge to x. In addition, x is globally asymptotically stable.

As of my third manuscript, I focused on providing some possible scenarios for general discrete competitive dynamical systems in the plane. I applied the results achieved to a class of second order difference equations of the form:

xn+1 = f(xn, xn-1), n = 0, 1, ...

where the function f(x; y) is decreasing in the variable x and increasing in the variable y. In my proofs I relied on a collection of well established theorems and results. Furthermore I illustrate my results with an application to equation:

xn+1 = x2n-1 / cx2n-1 + dxn + f, n = 0, 1, ... (3)

With initial conditions x-1 and xo arbitrary numbers and parameters c, d, f>0. Equation (3) is a special case of:

xn+1 - Cx2n-1 + Dxn +F / c2n-1 + dxn + f, n = 0, 1, ...

which of great interest to the field of difference equation and special cases of it considered by different scholars. It also turns out to be a non-linear perturbation of the Sigmoid Beverton-Holt model. I characterize completely the global bifurcations and dynamics equation. (3) with the basins of attraction of all its equilibria and periodic solutions. Moreover I provide techniques to investigates cases that are not covered by the established theorems in the theory of competitive maps.

Finally in my fourth manuscript I considered extending some existing theorems and proving some new global stability results, namely for difference equations that are of the form

xn+1 = f(xn, xn-1)

where f(x; y) is either increasing in the first and decreasing in the second variable, or decreasing in both variables. In addition I illustrate my results with examples and applications. I also provide a new proof for Pielou's equation (a mathematical model in population dynamics).

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