Reduction of continuous symmetries of chaotic flows by the method of slices

Details Reduction of continuous symmetries of chaotic flows by the method of slices Reduction of continuous symmetries of chaotic flows by the method of slices

2011-01-16

arxiv.org/abs/1101.3037v4

Nonlinear Sciences

Chaotic Dynamics

to appear in "Comm. Nonlinear Sci. and Numer. Simulat. (2011)" 12 pages, 8 figures

Scientific paper

10.1016/j.cnsns.2011.07.007

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a `slice' defined by minimizing the distance to a single generic `template' intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic / turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the template's local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO(2) symmetry of the complex Lorenz equations.

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