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...

Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

We explore the convergence/divergence of the normal form for a singularity of a vector field on ${ℂ}^n$ with nilpotent linear part. We show that a Gevrey-$\alpha$ vector field $X$ with a nilpotent linear part can be reduced to a normal form of Gevrey-$1+\alpha$ type with the use of a Gevrey-$1+\alpha$ transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

We establish a Poincaré-Dulac theorem for sequences ${\left(G_n\right)}_{n\in \mathbb{Z}}$ of holomorphic contractions whose differentials $d_0{G}_n$ split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of $ℂ{\mathbb{P}}^k$. In this context, our normalization result allows to estimate precisely the distortions of ellipsoids...

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field.

We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs $f:({ℂ}^2,0)⟶(ℂ,0)$ which take into account the inflection points of the fibres of $f$. We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.

We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${ℂ}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then...

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\alpha$-Gevrey vector field with an hyperbolic linear part admits a smooth $\beta$-Gevrey transformation to a smooth $\beta$-Gevrey normal form. The Gevrey order $\beta$ depends on...

We establish a polynomial normal form for a vector field having a limit cycle of multiplicity 2. The smooth classification problem for such fields is closely related to the problem of classification of germs $\Delta :({ℝ}^1,0)\to ({ℝ}^1,0)$, $\Delta \left(x\right)=x+c{x}^2+\cdots$, solved by F. Takens in 1973. Such germs appear as the germs of Poincaré return maps for semistable cycles, and a smooth conjugacy between any two such germs may be extended to a smooth orbital equivalence between the original fields.If one deals with smooth conjugacy of flows rather than...

Nous considérons les champs de vecteurs analytiques de $({ℂ}^n,0)$ de partie linéaire diagonale non nulle et dont les valeurs propres ${\lambda }_i\in ℂ$ vérifient des relations de résonances toutes engendrées par une seule relation $(r,\lambda )=0$ pour un certain vecteur $r\in {\mathbb{N}}^n$ non nul. Nous montrons que, dans un système de coordonnées locales holomorphes au voisinages de $0\in {ℂ}^n$, de tels champs de vecteurs se “mettent" sous une forme normale partielle, tout en exhibant des variétés invariantes, si l’on fait une hypothèse de petits diviseurs diophantiens....