Stickiness is a ubiquitous property of dynamical systems. However, recognizing whether an orbit is temporarily `stuck' (and therefore very nearly quasiperiodic) is hard to detect. Outlined in this thesis is an approach to quantifying stickiness in area-preserving maps based on a tool called recurrence plots that is not very commonly used. With the analyses presented herein it is shown that recurrence plot methods can give very close estimates to stickiness exponents that were previously calculated using Poincare recurrence and other methods. To capture the dynamics, RP methods require shorter data series than more conventional methods and are able to represent a more-global analysis of recurrence. A description of stickiness of the standard map for a wide array of parameter strengths is presented and a start at analyzing the standard nontwist map is presented. / text

We examine a class of planar area preserving mappings and give a geometric condition that guarantees the existence of homoclinic points. To be more precise, let $f,g:R \to R$ be $C^1$ functions with domain all of $R$. Let $F:R^2 \to R^2$ denote a horizontal reflection in the curve $x=-f(y)$, and let $G:R^2 \to R^2$ denote a vertical reflection in the curve $y=g(x)$. We consider maps of the form $T=G \circ F$ and show that a simple geometric condition on the fixed point sets of $F$ and $G$ leads to the existence of a homoclinic point for $T$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-17 14:22:35.72