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Topological techniques for characterization of patterns in differential equations

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dc.contributor.authorNeville, Rachel A., author
dc.contributor.authorShipman, Patrick, advisor
dc.contributor.authorAdams, Henry, committee member
dc.contributor.authorKrummel, Amber, committee member
dc.contributor.authorShonkwiler, Clayton, committee member
dc.date.accessioned2017-09-14T16:05:42Z
dc.date.available2017-09-14T16:05:42Z
dc.date.issued2017
dc.description.abstractComplex data can be challenging to untangle. Recent advances in computing capabilities has allowed for practical application of tools from algebraic topology, which have proven to be useful for qualitative and quantitative analysis of complex data. The primary tool in computational topology is persistent homology. It provides a valuable lens through which to study and characterize complex data arising as orbits of dynamical systems and solutions of PDEs. In some cases, this includes leveraging tools from machine learning to classify data based on topological characteristics. We see a unique pattern arising in the persistence diagram of a class of one-dimensional discrete dynamical systems--even in chaotic parameter regimes, and connect this to the dynamics of the system in Chapter 2. Geometric pattern structure tell us something about the parameters driving the dynamics in the system as is the case for anisotropic Kuramoto-Sivashinsky equation which displays chaotic bubbling. We will see this in Chapters 3 and 4. Defects in pattern-forming systems be detected and tracked and studied to characterize the degree of order in near-hexagonal nanodot structures formed by ion bombardment, which will be developed in Chapter 5.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierNeville_colostate_0053A_14351.pdf
dc.identifier.urihttps://hdl.handle.net/10217/183981
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2000-2019
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectdifferential equations
dc.subjectpersistent homology
dc.subjectcomputational topology
dc.subjecttopological data analysis
dc.subjectpatterns
dc.titleTopological techniques for characterization of patterns in differential equations
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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