Date of Award

2015

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Orlando Merino

Abstract

In order to study the global behavior of difference equations, it is necessary to understand the local behavior in a neighborhood of a equilibrium point of the difference equation. This thesis focuses on two aspects of the local behavior of planar difference equations: the asymptotic behavior of a solution converging to a hyperbolic fixed point, and the local qualitative behavior of a non isolated fixed point whose jacobian matrix has a particular structure.

Manuscript 2 describes how closely a convergent solution {Xn} of (real or complex) di_ffrence equations xn+1 = J xn + fn(xn) can be approximated by its linearization zn+1 = J zn in a neighborhood of a fixed point; where xn is a m- vector, J is a constant m x m matrix and fn(y) is a vector valued function which is continuous in y for fixed n, and where fn(y) is small in a sense.

Manuscript 3 describes completely the local qualitative behavior of a real planar map in a neighborhood of a non-isolated fixed point whose jacobian matrix is similar to ( 1 10 1 ), also called a non-isolated 1-1 resonant fixed point. Theorem 3 gives conditions for four non-conjugate dynamical scenarios to occur.

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