Willis, A.P., Short, K.Y. and Cvitanovic, P. (2016) Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Physical Review E, 93. 022204 . ISSN 1550-2376
Abstract
Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e., for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied one-dimensional PDEs, to a spatially three-dimensional fluid flow, and show that it is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 American Physical Society. This is an author-produced version of a paper subsequently published in Physical Review E. Uploaded in accordance with the publisher's self-archiving policy |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Funding Information: | Funder Grant number ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL (EPSRC) EP/K03636X/1 |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 21 Jan 2016 11:05 |
Last Modified: | 09 Mar 2016 22:08 |
Published Version: | http://dx.doi.org/10.1103/PhysRevE.93.022204 |
Status: | Published |
Publisher: | American Physical Society |
Refereed: | Yes |
Identification Number: | 10.1103/PhysRevE.93.022204 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:93986 |