Bifurcation Phenomena in Two-Dimensional Piecewise Smooth Discontinuous Maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border, and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, many switching dynamical systems have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. The theory for understanding the bifurcation phenomena in such systems is not available yet. In this paper we present the first approach to the problem of analysing and classifying the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We explain the bifurcations occurring in the static VAR compensator used in electrical power systems, using the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps

On the physical and chemical dynamics of chromatin

The research performed leading to this dissertation is an endeavor to explore two broad classes of developmental phenomena in the chromatin complex in eukaryotic cells --- physical, for instance, long range interactions between enhancers and promoters, and chemical, such as epigenetic chromatin silencing. I begin by introducing the reader to both types of phenomena, and then set the stage for our strategy in the exploration of the physical side of these processes by creating a new machinery from existing pieces of polymer physics. I then make a brief foray into theoretical realms in an attempt to answer the question of what kinds of conformations of polymers dominate in what regimes. Subsequently, I proceed to consider the problem of analyzing and interpreting data from a major technique of probing the behavior of the chromatin complex in vivo --- Chromosome Conformation Capture --- towards which end we have developed and implemented a new and robust algorithm called `G.R.O.M.A.T.I.N.'. Subsequently, I explore how similar ideas may be invoked in the analysis of direct microscopic observations of native chromatin structure via Fluorescence in situ Hybridization. Following this, I look at the problems of epigenetic chromatin silencing domain formation and stability in the presence of titration feedback and of stochastic noise, and demonstrate how the widely accepted polymerization model of silencing is consistent with Chromatin Immunoprecipitation data from silencing domains in budding yeast. I finally conclude with musings on recent evidence pinpointing the need to unify the physical and chemical pictures into one grand formulation.Ph. D.Includes bibliographical referencesIncludes vitaby Manjul Aprati