Patterns in dynamics

In this paper we introduce and explore the idea of persistent homology (PH) and discuss several applications of this computational topology tool beyond its intended purpose. In particular we apply persistence to data generated by dynamical systems. The application of persistent homology to the circle map will lead us to rediscover the well-known result about the distribution of points in the orbit of this ergodic system called the Three Distance Theorem. We then apply PH to data extracted from several models of ion bombardment of a solid surface. This will present us with an opportunity to discuss new ways of interpreting PH data by introducing statistics on its output. Using these statistics we will begin to develop a technique to answer questions of interest to physicists about the degree of ordering present in the topography of a solid surface after ion bombardment. Finally we observe some inherent limitations in PH and, through simple examples, develop techniques to improve the technology. Specically we will implement algorithms to iteratively spread points on a real algebraic variety and demonstrate that the methodology works to improve the signals in the output of PH.

Name thesis:degree_name

Master of Science (M.S.)

Level thesis:degree_level

Masters

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

Colorado State University. Libraries

Year dc:date.issued

2012

Authors dc:creator

• Motta, Francis Charles, author
• Shipman, Patrick D., advisor
• Bradley, R. Mark, committee member
• Cavalieri, Renzo, committee member
• Dangelmayr, Gerhard, committee member

• dynamical systems
• ion bombardment
• topology
• patterns
• persistent homology

Statement dc:rights

• Copyright 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.

Language dc:language.iso

eng, English

Identifier URI

https://doi.org/10.25675/3.018396

OAI identifier oai:identifier

oai:mountainscholar.org:10217/88357