Title:
Topics in dynamical systems

dc.contributor.advisor Bunimovich, Leonid
dc.contributor.advisor Webb, Benjamin
dc.contributor.author Shu, Longmei
dc.contributor.committeeMember De La Llave, Rafael
dc.contributor.committeeMember Heitsch, Christine
dc.contributor.committeeMember Iliev, Plamen
dc.contributor.committeeMember Alexeev, Alexander
dc.contributor.department Mathematics
dc.date.accessioned 2019-08-21T13:52:36Z
dc.date.available 2019-08-21T13:52:36Z
dc.date.created 2019-08
dc.date.issued 2019-06-14
dc.date.submitted August 2019
dc.date.updated 2019-08-21T13:52:36Z
dc.description.abstract The thesis consists of two parts. the first one is dealing with isosspectral transformations and the second one with the phenomenon of local immunodeficiency. Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors. Chapter 1 analyzes what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks. In the recently developed theory of isospectral transformations of networks isospectral compressions are performed with respect to some chosen characteristics (attributes) of the network's nodes (edges). Each isospectral compression (when a certain characteristic is fixed) defines a dynamical system on the space of all networks. Chapter 2 shows that any orbit of this dynamical system which starts at any finite network (as the initial point of this orbit) converges to an attractor. This attractor is a smaller network where the chosen characteristic has the same value for all nodes (or edges). We demonstrate that isospectral compressions of one and the same network defined by different characteristics of nodes (or edges) may converge to the same as well as to different attractors. It is also shown that a collection of networks may be spectrally equivalent with respect to some network characteristic but nonequivalent with respect to another. These results suggest a new constructive approach which allows to analyze and compare the topologies of different networks. Some basic aspects of the recently discovered phenomenon of local immunodeficiency generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated in chapter 3. We prove that stable with respect to perturbations local immunodeficiency (LI) already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large, but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topologies. In particular, the scale-free property of a CR network, assumed previously, is not required.
dc.description.degree Ph.D.
dc.format.mimetype application/pdf
dc.identifier.uri http://hdl.handle.net/1853/61746
dc.language.iso en_US
dc.publisher Georgia Institute of Technology
dc.subject Isospectral transformations
dc.subject Generalized eigenvectors
dc.subject Spectral equivalence
dc.subject Attractors
dc.subject Cross-immunoreactivity network
dc.subject Local immunodeficiency
dc.subject Minimal stable network
dc.title Topics in dynamical systems
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
thesis.degree.level Doctoral
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