The infinite level normal forms for non-resonant double Hopf singularities

In this paper, we explore hypernormal forms of vector fields that have non-resonant double Hopf singularities with a non-zero radial cubic part. Our primary focus is on investigating the infinite-level normal form classification of this type of singularities. We provide a normal form decomposition in terms of planar-rotating and planar-radial vector fields, which greatly facilitate the pattern recognition and analysis of the corresponding generalized homological maps. Notably, our paper represents the first instance of the normal form classification for general non-resonant double Hopf singularities without structural symmetry.