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1	Quantifying stickiness in 2D area-preserving maps by means of recurrence plots Eschbacher, Peter Andrew 03 September 2009 (has links) 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 Physics Area-Preserving Maps Stickiness Recurrence Plots
2	Homoclinic Points in the Composition of Two Reflections Jensen, ERIK 17 September 2013 (has links) 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 Area preserving maps Homoclinic points Mathematics Dynamical systems
3	The Lie Symmetries of a Few Classes of Harmonic Functions Petersen, Willis L. 23 May 2005 (has links) (PDF) In a differential geometry setting, we can analyze the solutions to systems of differential equations in such a way as to allow us to derive entire classes of solutions from any given solution. This process involves calculating the Lie symmetries of a system of equations and looking at the resulting transformations. In this paper we will give a general background of the theory necessary to develop the ideas of working in the jet space of a given system of equations, applying this theory to harmonic functions in the complex plane. We will consider harmonic functions in general, harmonic functions with constant Jacobian, harmonic functions with fixed convexity and a few other subclasses of harmonic functions. Lie symmetries harmonic functions area preserving Lie group geometric function theory schlicht functions differential geometry manifolds Lie algebra Mathematics
4	Estabilidade de variações que preservam áreas em formas espaciais. / Stability of área-preserving variations in space forms. Nascimento, Arlyson Alves do 23 April 2009 (has links) In this dissertation, we deal with compact hypersurfaces without boundary immersed in space forms such that Sr+1/S1 is constant. They are critical points for an area-preserving variational problem. We show that they are r-stable if and only if they are totally umbilical hypersurfaces. / Fundação de Amparo a Pesquisa do Estado de Alagoas / O objetivo desta dissertação é estudar as hipersuperfícies compactas sem bordo e imersas em formas espaciais com Sr+1 / S1 constante, onde Sr+1 é a (r + 1)-ésima função simétrica das curvaturas principais. Tais hipersuperfícies são os pontos críticos de um problema variacional que preserva área. Demonstraremos que tais imersões são r-estáveis se, e somente se, elas forem hipersuperfícies totalmente umbílicas. R-ésima curvatura média R-estáveis Variações que preservam área Rth mean curvatures R-stability Area-preserving variation
5	On Visualizing Branched Surface: an Angle/Area Preserving Approach Zhu, Lei 12 September 2004 (has links) The techniques of surface deformation and mapping are useful tools for the visualization of medical surfaces, especially for highly undulated or branched surfaces. In this thesis, two algorithms are presented for flattened visualizations of multi-branched medical surfaces, such as vessels. The first algorithm is an angle preserving approach, which is based on conformal analysis. The mapping function is obtained by minimizing two Dirichlet functionals. On a triangulated representation of vessel surfaces, this algorithm can be implemented efficiently using a finite element method. The second algorithm adjusts the result from conformal mapping to produce a flattened representation of the original surface while preserving areas. It employs the theory of optimal mass transport via a gradient descent approach. A new class of image morphing algorithms is also considered based on the theory of optimal mass transport. The mass moving energy functional is revised by adding an intensity penalizing term, in order to reduce the undesired "fading" effects. It is a parameter free approach. This technique has been applied on several natural and medical images to generate in-between image sequences. Algorithms Area-preserving flattening Conformal mapping Image interpolation Interpolation Monge-Kantorovich problem Surface flattening Surfaces (Technology) Analysis Surfaces (Technology) Analysis Free radicals (Chemistry) Organosulfur compounds Oxidation Algorithms Conformal mapping Interpolation Chemical kinetics Dimethyl sulfide Oxidation

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