Repository logo
 

Persistent homology of the logistic map: an exploration of chaos

Simple item page
dc.contributor.authorNeville, Rachel, author
dc.contributor.authorShipman, Patrick, advisor
dc.contributor.authorPeterson, Chris, committee member
dc.contributor.authorThompson, Stephen, committee member
dc.date.accessioned2007-01-03T06:43:00Z
dc.date.available2007-01-03T06:43:00Z
dc.date.issued2014
dc.description.abstractGiven a discrete sampling of points, how can one reconstruct the underlying geometric object? Further, the question arises how can one discern between noise and sampling distortion and important topological features. Algebraic and topological techniques used computationally can prove to be powerful and currently unconventional tools to understand the "shape" of data. In recent years, persistent homology has been explored as a computational way to capture information regarding the longevity of topological features of discrete data sets. In this project, the persistent homology of functions is explored specifically as a way of examining features of functions. Persistent homology tracks the longevity of connected components of level sets in a persistence diagram. By connecting points generated by a discrete time dynamical system with line segments, this data can be viewed as a (piece-wise linear) function, persistent homology is used to track features of the data. This provides a novel and useful tool for computationally examining dynamical systems. The logistic map is one of the simplest examples of a nonlinear map that displays periodic behavior for some parameter values, but for others, displays chaotic behavior. When the persistence diagram is generated for an orbit of the logistic map, all of the points surprisingly lie approximately on a line. This is not true for a general sequence. This pattern arises not only after stability has been reached in the periodic case, but also as points approach stability for parameters in the periodic regime but also perhaps more surprisingly, for parameter values that lie in the chaotic regime as well. In fact, the slope of this line is fairly similar as the parameter values are varied. This arises from the order in which the points pair to form the persistence diagram and a scaling factor seen in the periodic regime of a class of maps (including the logistic map). It is interesting that the effects of this scaling are still seen in the chaotic regime. This pattern not only arises for the logistic map, but for other unimodal maps and other higher dimensional systems that are "close" to these maps such as the Lorenz system.
dc.format.mediumborn digital
dc.format.mediummasters theses
dc.identifierNeville_colostate_0053N_12329.pdf
dc.identifier.urihttp://hdl.handle.net/10217/82626
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.subjectchaos
dc.subjectunimodal
dc.subjectpersistent homology
dc.subjectlogistic map
dc.titlePersistent homology of the logistic map: an exploration of chaos
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.levelMasters
thesis.degree.nameMaster of Science (M.S.)

Files

Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
Neville_colostate_0053N_12329.pdf
Size:
2.51 MB
Format:
Adobe Portable Document Format
Description:
Download

Collections

2000-2019
Theses and Dissertations