Title:
Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems

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dc.contributor.advisor Bunimovich, Leonid
dc.contributor.author Webb, Benjamin Zachary en_US
dc.contributor.committeeMember Bakhtin, Yuri
dc.contributor.committeeMember Dieci, Luca
dc.contributor.committeeMember Randall, Dana
dc.contributor.committeeMember Weiss, Howie
dc.contributor.department Mathematics en_US
dc.date.accessioned 2011-07-06T16:25:08Z
dc.date.available 2011-07-06T16:25:08Z
dc.date.issued 2011-03-18 en_US
dc.description.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 one-dimensional 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 time-delayed 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 one-dimensional 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. en_US
dc.description.degree Ph.D. en_US
dc.identifier.uri http://hdl.handle.net/1853/39521
dc.publisher Georgia Institute of Technology en_US
dc.subject Schwarzian derivative en_US
dc.subject Global stability en_US
dc.subject Dynamical networks en_US
dc.subject Spectral equivalence en_US
dc.subject Graph transformations en_US
dc.subject.lcsh Complex matrices
dc.subject.lcsh Attractors (Mathematics)
dc.subject.lcsh Eigenvalues
dc.title Isospectral graph reductions, estimates of matrices' spectra, and eventually negative Schwarzian systems en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.advisor Bunimovich, Leonid
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isAdvisorOfPublication 8385d52f-b627-4839-8603-2683ea2daa55
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
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