# 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 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 × 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 ([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.

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