In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of ${ℂ}^n$, fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part $S$ which ensures that if such a perturbation of $S$ is formally conjugate to $S$ then it is also holomorphically conjugate to it. We study the normal form problem relatively to $S$. We give a condition on $S$ that ensures that there...

We implement a singularity theory approach, the path formulation, to classify ${𝔻}_3$-equivariant bifurcation problems of corank 2, with one or two distinguished parameters, and their perturbations. The bifurcation diagrams are identified with sections over paths in the parameter space of a ${𝔻}_3$-miniversal unfolding $F_0$ of their cores. Equivalence between paths is given by diffeomorphisms liftable over the projection from the zero-set of $F_0$ onto its unfolding parameter space. We apply our results to degenerate...