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

TL;DR

This paper introduces a visual framework for understanding high-dimensional chaotic flows by using symmetry reduction techniques to create an atlas of the state space, revealing the structure of solutions and their role in chaos.

Contribution

It develops a method to visualize and analyze the symmetry-reduced state space of chaotic flows using charts and slices, facilitating understanding of solution organization.

Findings

Charts reveal the organization of solutions in the reduced state space

Symmetry reduction simplifies the identification of equilibria and periodic orbits

Visualization of unstable manifolds clarifies their role in turbulence

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.