  In this paper, we deal with hypernormal forms of non-resonant double Hopf
singularities. We investigate the infinite level normal form classification of
such singularities with nonzero radial cubic part. We provide a normal form
decomposition of normal form vector fields in terms of planar-rotating and
planar-radial vector fields. These facilitate the pattern recognition and
analysis of the corresponding generalized homological maps. This paper is the
first instance of the normal form classification for generic non-resonant
double Hopf singularities without structural symmetry.
