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Willis AP, Short KY, Cvitanović P. Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Phys Rev E. 2016;93(2):022204doi: 10.1103/PhysRevE.93.022204.
Willis, A. P., Short, K. Y., & Cvitanović, P. (2016). Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Physical review. E, 93(2), 022204. https://doi.org/10.1103/PhysRevE.93.022204
Willis, Ashley P, et al. "Symmetry reduction in high dimensions, illustrated in a turbulent pipe." Physical review. E vol. 93,2 (2016): 022204. doi: https://doi.org/10.1103/PhysRevE.93.022204
Willis AP, Short KY, Cvitanović P. Symmetry reduction in high dimensions, illustrated in a turbulent pipe. Phys Rev E. 2016 Feb;93(2):022204. doi: 10.1103/PhysRevE.93.022204. Epub 2016 Feb 05. PMID: 26986328.
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Phys Rev E. 2016 Feb;93(2):022204. doi: 10.1103/PhysRevE.93.022204. Epub 2016 Feb 05.

Symmetry reduction in high dimensions, illustrated in a turbulent pipe.

Physical review. E

Ashley P Willis, Kimberly Y Short, Predrag Cvitanović

Affiliations

  1. School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, United Kingdom.
  2. Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA.

PMID: 26986328 DOI: 10.1103/PhysRevE.93.022204

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.

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