Symmetries of center singularities of plane vector fields

Details Symmetries of center singularities of plane vector fields Symmetries of center singularities of plane vector fields

2009-07-02

arxiv.org/abs/0907.0359v1

Mathematics

Dynamical Systems

30 pages, 4 figures, submitted

Scientific paper

Let D be a closed unit 2-disk on the plane centered at the origin 0, and F be a smooth vector field on D such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus topologically O is a ``center'' singularity. Let p:D\0-->(0,+\infty) be the function associating to each $z\not=O$ its period with respect to F. This function can be discontinuous at O. Let Diff(F) be the group of all diffeomorphisms of D which preserve orientation and orbits of F. We prove that p smoothly extends to all of D if and only if the 1-jet of F at the origin is a non-degenerate linear map, and that in this case Diff(F) is homotopy equivalent to the circle.

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