Abstract

This dissertation investigates the dynamics of some second-order difference equations and systems of difference equations whose defining functions satisfy certain monotonicity properties. In each study we utilize the theory for specific classes of monotone difference equations to establish local and global dynamics. Manuscript 1 is an introduction that provides fundamental definitions and important results for difference equations that are used throughout the rest of the thesis. Manuscript 2 presents some potential global dynamic scenarios for competitive systems of difference equations in the plane. These results are extended to apply to the class of second-order difference equations whose transition functions are decreasing in the first variable and increasing in the second. In particular, these results are applied to investigate the following equation as a case study: [Mathematical equations cannot be displayed here, refer to PDF] (1) where the initial conditions χ-1 and χ0 are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C, E, a, d, f ≥ 0, C+E > 0, a + C > 0, and a + d > 0. A rich collection of additional dynamical behaviors for Equation (1) are established to provide a nearly complete characterization of its global dynamics with the basins of attraction of equilibria and periodic solutions. [Mathematical equations cannot be displayed here, refer to PDF] (2) Here f is a continuous function nondecreasing in both arguments, the parameter a is a positive real number, and the initial conditions χ-1 and χ0 are arbitrary nonnegative numbers such that the solution is defined. Local and global dynamics of Equation (2) are presented in the event f is chosen to be a certain type of linear or quadratic polynomial. Particular consideration is given to the existence problem of period-two solutions. Manuscript 4 presents an order-k generalization of Equation (2), [Mathematical equations cannot be displayed here, refer to PDF] (3) where f remains a function nondecreasing in all of its arguments, a > 0, and χ0, χ1,: : : , χ1-k ≥ 0. We examine several interesting examples in which f is a transcendental function. This manuscript establishes conditions under which Equation (3) possesses a unique positive equilibrium that is a global attractor of all solutions with positive initial conditions. In particular, results are presented for the special case in which f(x, : : : , x) is chosen to be a concave function.

Highlights

  • 1.1 Second-Order Difference EquationsDiscrete dynamical systems describe the evolution of a quantity or population whose changes are measured over discrete time intervals

  • The paramount goal is to determine the global dynamics of a difference equation by analytically characterizing the end behavior of all solutions as n → ∞

  • Corollary 2 If the nonnegative cone of is a generalized quadrant in Rn, and if T has no fixed points in u1, u2 other than u1 and u2, the interior of u1, u2 is either a subset of the basin of attraction of u1 or a subset of the basin of attraction of u2. These results have been utilized in papers such as [1] to determine the basins of attraction of certain fixed points; they provide a theoretical foundation for the investigation of the dynamics of competitive and cooperative systems

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Summary

Second-Order Difference Equations

Discrete dynamical systems describe the evolution of a quantity or population whose changes are measured over discrete time intervals. Difference equations may be thought of as recurrence relations that describe a discrete dynamical system by relating the size of the state (or generation), often denoted xn+1, to some function of the sizes of several past states xn, xn−1, . . For example, a second-order autonomous difference equation may take the form xn+1 = f (xn, xn−1), n = 0, 1, . For each choice of initial conditions, Equation (1) has a unique solution {xn}∞ n=−1. Much initial investigation in this field of research focuses on describing the local dynamics of such difference equations by examining the short-term trajectory of solutions for different choices of initial conditions. The paramount goal is to determine the global dynamics of a difference equation by analytically characterizing the end behavior of all solutions as n → ∞

Local Stability Analysis
Monotone Systems of Difference Equations
Introduction
Preliminaries
Main Results
Equilibrium Solutions of
Local Stability Analysis of the Equilibrium Solutions
Global Dynamics of Equation (1)
Introduction and Preliminaries
Local Stability
Translated Linear-Linear: f (u, v) = cu + dv + k
Quadratic-Linear: f (u, v) = cu2 + dv
Linear-Quadratic: f (u, v) = cu + dv2
General Stability Results and Global Attractivity
Local Stability of an Equilibrium The linearized equation of
Existence and Global Attractivity of a Unique Positive Equilibrium
Exponential: f (u, v) = p(1 − e−u) + q(1 − e−v)
Inverse Tangent: f (u, v) = p arctan(u) + q arctan(v)
Trigonometric: f (u, v) = p (u + sin(u)) + q (v + sin(v))
Order-k Linear
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