Symmetry Reduction by Lifting for Maps

Details Symmetry Reduction by Lifting for Maps Symmetry Reduction by Lifting for Maps

2011-11-16

arxiv.org/abs/1111.3887v2

Nonlinear Sciences

Chaotic Dynamics

laTeX, 31 pages, 5 figures

Scientific paper

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincar\'{e} section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.

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