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Topological techniques for characterization of patterns in differential equations




Neville, Rachel A., author
Shipman, Patrick, advisor
Adams, Henry, committee member
Krummel, Amber, committee member
Shonkwiler, Clayton, committee member

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Complex 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.


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differential equations
persistent homology
computational topology
topological data analysis


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