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Blowing up the zero section for "Chasse au Canard" (some new kind of geometric canards)

In this paper "Canard cycles and center manifolds" one encounters the blowing up of a non isolated set or manifold of singularities of a vector field or a singular foliation. This is a ...
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Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow

However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
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