Cartography of high-dimensional flows: A visual guide to sections and slices

Abstract

Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a `template'. Together these charts provide an atlas of the symmetry-reduced `slice' of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits. Visualizations of these solutions and their unstable manifolds reveal their interrelations and the role they play in organizing turbulence/chaos.

12 Pages, 12 figures

References in corpus (5)

• The Gribov problem and QCD dynamics
• Relative periodic orbits in transitional pipe flow
• Highly-symmetric travelling waves in pipe flow
• Continuous symmetry reduction and return maps for high-dimensional flows
• Reduction of continuous symmetries of chaotic flows by the method of slices

Cited by in corpus (11)

• Relative periodic orbits form the backbone of turbulent pipe flow
• Revealing the state space of turbulent pipe flow by symmetry reduction
• Analysis of Kolmogorov Flow and Rayleigh-Bénard Convection using Persistent Homology
• Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems
• Reduction of SO(2) symmetry for spatially extended dynamical systems
• Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto-Sivashinsky system
• Periodic orbit analysis of a system with continuous symmetry - a tutorial
• Symmetry related dynamics in parallel shear flows
• Complexity of the laminar-turbulent boundary in pipe flow
• Hopf fibrations for turbulent pipe flows
• Symmetry reduction of turbulent pipe flows