#### 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 {X_{n}} of (real or complex) difference equations x_{n+1} = *J* x* _{n}* +

*f*

*(x*

_{n}*) can be approximated by its linearization*

_{n}*z*

_{n}_{+1}=

*J*

*z*in a neighborhood of a fixed point; where x

_{n}*is a m- vector,*

_{n}*J*is a constant m x m matrix and

*f*

*(γ) is a vector valued function which is continuous in γ for fixed*

_{n}*n*, and where

*f*

*(γ) is small in a sense.*

_{ n}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.

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