Reduction of continuous symmetries of chaotic flows by the method of slices
arXiv:1101.3037 · doi:10.1016/j.cnsns.2011.07.007
Abstract
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.
to appear in "Comm. Nonlinear Sci. and Numer. Simulat. (2011)" 12 pages, 8 figures
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- Model order reduction for stochastic dynamical systems with continuous symmetries
- Hopf fibrations for turbulent pipe flows
- Symmetry reduction of turbulent pipe flows
- Cartography of high-dimensional flows: A visual guide to sections and slices
- Analytical and numerical study of the hidden boundary of practical stability: complex versus real Lorenz systems