Symmetry reduction in high dimensions, illustrated in a turbulent pipe

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 1-dimensional PDEs, to a spatially 3-dimensional fluid flow, and show that 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.

arXiv comment5 pages, 4 figures, 1 table

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• Intermittent direction reversals of moving spatially-localized turbulence observed in two-dimensional Kolmogorov flow
• Sensitivity of long periodic orbits of chaotic systems
• Multi-scale invariant solutions in plane Couette flow: a reduced-order model approach
• Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow
• Analysis and modeling of localized invariant solutions in pipe flow
• New equilibrium solution branches of plane Couette flow discovered using a project-then-search method
• Equilibria, periodic orbits and computing them