Continuous symmetry reduction and return maps for high-dimensional flows

Abstract

We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In the method of moving frames (or method of slices) the state space is sliced locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point. In either approach, numerical computations can be performed in the original state-space representation, and the solutions are then projected onto the symmetry-reduced state space. The two methods are illustrated by reduction of the complex Lorenz system, a 5-dimensional dissipative flow with rotational symmetry. While the Hilbert polynomial basis approach appears unfeasible for high-dimensional flows, symmetry reduction by the method of moving frames offers hope.

32 pages, 7 figures

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• Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto-Sivashinsky system
• Reduction of continuous symmetries of chaotic flows by the method of slices
• Periodic orbit analysis of a system with continuous symmetry - a tutorial
• 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
• The morphology of MSS-sequences in a wide class of unimodal maps, its structure and decomposition