#### Date of Award

2015

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### 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.

#### Recommended Citation

Jamieson, William T., "PLANAR DIFFERENCE EQUATIONS: ASYMPTOTIC BEHAVIOR OF SOLUTIONS AND 1-1 RESONANT POINTS" (2015). *Open Access Dissertations.* Paper 336.

http://digitalcommons.uri.edu/oa_diss/336

#### Terms of Use

All rights reserved under copyright.