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Dissertation
Analysis of a strange attractor in R⁴
Doctor of Philosophy (Ph.D.), Drexel University
Dec 2011
DOI:
https://doi.org/10.17918/etd-4054
Abstract
Strange attractors in R³ are remarkably well understood because they may be classified through a topological analysis. This involves determining the organization of the unstable periodic orbits in the attractor by computing linking numbers for pairs of these orbits and then identifying a branched manifold that supports these and all other orbits, and serves to identify the mechanism that generates chaotic behavior. This topological invariant can be calculated in R³ but there is no analog for higher dimensions. We analyze a strange attractor generated by a four dimensional dynamical system and show the first steps in extending the current topological analysis program to higher dimensions. The linking numbers for pairs of unstable periodic orbits are computed in three different ways. Firstly, through projections of the attractor to three dimensional subspaces. Secondly, using a recently proposed higher dimensional linking integral to compute linking numbers in R⁴ for the first time. Thirdly, with a dimensionality reduction technique, Locally Linear Embedding, that successfully represents pairs of orbits in three dimensions allowing the organization within the strange attractor to be determined.
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Details
- Title
- Analysis of a strange attractor in R⁴
- Creators
- Benjamin Coy - DU
- Contributors
- Robert Gilmore (Advisor) - Drexel University (1970-)
- Awarding Institution
- Drexel University
- Degree Awarded
- Doctor of Philosophy (Ph.D.)
- Publisher
- Drexel University; Philadelphia, Pennsylvania
- Resource Type
- Dissertation
- Language
- English
- Academic Unit
- College of Arts and Sciences; Physics; Drexel University
- Other Identifier
- 4054; 991014632596004721