Codimension-three unfoldings of reflectionally symmetric planar vector fields

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Abstract
We consider unfoldings of a codimension-three singularity of reflectionally symmetric planar vector fields that provide a link between the different codimension-two unfoldings known in the literature. This work has been motivated by and can be applied to the study of bifurcations at infinity of planar vector fields with an even rotational symmetry. The codimension-three singularity is determined by its 4-jet, which gives eight classes. We analyse the 4-jet of the unfoldings, of which there are 22 classes. The unfoldings are (conjecturally) versal for 16 classes. The other six classes of unfoldings contain centres, and it is necessary to consider a jet of higher order, which is only indicated here. In our analysis we use that the 4-jet can be reduced to a quadratic vector field.

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