Title:  Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems 
Author:  Webb, Benjamin Zachary 
Abstract:  This dissertation can be essentially divided into two parts. The first, consisting of Chapters I, II, and III, studies the graph theoretic nature of complex systems. This includes the spectral properties of such systems and in particular their influence on the systems dynamics. In the second part of this dissertation, or Chapter IV, we consider a new class of onedimensional dynamical systems or functions with an eventual negative Schwarzian derivative motivated by some maps arising in neuroscience. To aid in understanding the interplay between the graph structure of a network and its dynamics we first introduce the concept of an isospectral graph reduction in Chapter I. Mathematically, an isospectral graph transformation is a graph operation (equivalently matrix operation) that modifies the structure of a graph while preserving the eigenvalues of the graphs weighted adjacency matrix. Because of their properties such reductions can be used to study graphs (networks) modulo any specific graph structure e.g. cycles of length n, cliques of size k, nodes of minimal/maximal degree, centrality, betweenness, etc. The theory of isospectral graph reductions has also lead to improvements in the general theory of eigenvalue approximation. Specifically, such reductions can be used to improved the classical eigenvalue estimates of Gershgorin, Brauer, Brualdi, and Varga for a complex valued matrix. The details of these specific results are found in Chapter II. The theory of isospectral graph transformations is then used in Chapter III to study timedelayed dynamical systems and develop the notion of a dynamical network expansion and reduction which can be used to determine whether a network of interacting dynamical systems has a unique global attractor. In Chapter IV we consider onedimensional dynamical systems of an interval. In the study of such systems it is often assumed that the functions involved have a negative Schwarzian derivative. Here we consider a generalization of this condition. Specifically, we consider the functions which have some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class of functions. This includes both systems with regular as well as chaotic dynamic properties. 
Type:  Dissertation 
URI:  http://hdl.handle.net/1853/39521 
Date:  20110318 
Publisher:  Georgia Institute of Technology 
Subject: 
Schwarzian derivative
Global stability Dynamical networks Spectral equivalence Graph transformations Complex matrices Attractors (Mathematics) Eigenvalues 
Department:  Mathematics 
Advisor:  Committee Chair: Bunimovich, Leonid; Committee Member: Bakhtin, Yuri; Committee Member: Dieci, Luca; Committee Member: Randall, Dana; Committee Member: Weiss, Howie 
Degree:  Ph.D. 
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