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Symmetry reduction by the method of slices quotients the continuoussymmetries of chaotic flows by replacing the original state space by a set ofcharts, each covering a neighborhood of a dynamically important class ofsolutions, qualitatively captured by a `template'. Together these chartsprovide an atlas of the symmetry-reduced `slice' of state space, charting theregions of the manifold explored by the trajectories of interest. Within theslice, relative equilibria reduce to equilibria and relative periodic orbitsreduce to periodic orbits. Visualizations of these solutions and their unstablemanifolds reveal their interrelations and the role they play in organizingturbulence/chaos.

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