# Planar difference equations: Asymptotic behavior of solutions and 1-1 resonant points

#### 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}*fn*(

*x*) can be approximated by its linearization

_{n}*z*

^{n}_{+1}= J

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

_{n}*x*is a

_{n}*m*-vector,

*J*is a constant

*m*×

*m*matrix and

*f*(

^{n}*y*) is a vector valued function which is continuous in

*y*for fixed

*n,*and where

*f*(

_{n}*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 (

**[special characters omitted]**), also called a non-isolated

*1-1 resonant*fixed point. Theorem 3 gives conditions for four non-conjugate dynamical scenarios to occur.^

#### Subject Area

Mathematics

#### Recommended Citation

William T Jamieson,
"Planar difference equations: Asymptotic behavior of solutions and 1-1 resonant points"
(2015).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3689186.

http://digitalcommons.uri.edu/dissertations/AAI3689186