We consider a planar vector field X near a saddle type p : -q resonant singular point. Assuming that it has a normal form with a Gevrey-d expansion (like d = p + q which is in particular the case when starting from an analytic vector field) we show that X can be linearized working with a change of coordinates that is of Gevrey order d in certain log-like variables, called compensators or also tags, multiplied by the first integral u = x(q)y(P) of the linear part. Next we consider the unfolding of such a resonance, and provide (weaker) Gevrey-type linearization using compensators.

Poincare Dulac normal form; conjugacy; normal form; Mourtada type function; tag monomial Gevrey asymptotic

We show that any germ of smooth hyperbolic diffeomophism at a fixed point is conjugate to its linear part, using a transformation with a Mourtada type functions, which (roughly) means that it may contain terms like x log vertical bar x vertical bar. Such a conjugacy admits a Mourtada type expansion. In the planar case, when the fixed point is a p : -q resonant saddle, and if we assume that the diffeomorphism is of Gevrey class, we give an upper bound on the Gevrey estimates for this expansion.

We prove a partial linearization theorem for planar nonautonomous differential equations with one center-like direction and one hyperbolic direction. Gap conditions are formulated in terms of the dichotomy spectral intervals. The result is applied to the Duffing–van der Pol oscillator under a nonautonomous parametric perturbation.

Given a 1:−1 resonant saddle singularity of a planar analytic vector field, we provide a linearization procedure using a series expansion in compensators of Mourtada-type, and show that this series has Gevrey-1 asymptotics. In case of an analytic Poincaré–Dulac normal form we show that this transformation is analytic as a function of the compensators.

Let \({X = \lambda_{1} x_{1}{\frac{\partial}{\partial {x_1}}} + \lambda_2 x_2 \frac{\partial}{\partial {x_2}} + O(|x|^2)}\) be an analytic vector field near x = 0. We suppose that the linear part of this vector field has real eigenvalues λ1, λ2 and that the ratio \({\eta = -\frac{\lambda_1}{\lambda_2}}\) is a positive irrational number. In a previous paper of the first author and P. De Maesschalck, it was shown that any analytic saddle can be conjugated analytically to a form ‘as close as desired’ to the formal normal form. In this paper we will iterate and renormalize these conjugacies. The iteration of this procedure will be strongly connected to the diophantine properties of η and we will establish the convergence of this process. A consequence of this convergence will be the two dimensional version of the by now classical linearization theorem of Bruno.

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 provide smooth local normal forms near singularities that appear in planar singular perturbation problems after application of the well-known family blow up technique. The local normal forms preserve the structure that is provided by the blow-up transformation. In a similar context, C k -structure-preserving normal forms were shown to exist, for any finite k. In this paper, we improve the smoothness by showing the existence of a C ∞ normalizing transformation, or in other cases by showing the existence of a single normalizing transformation that is C k for each k, provided one restricts the singular parameter ε to a (k-dependent) sufficiently small neighborhood of the origin.

We study local analytic simplification of families of analytic maps near a hyperbolic fixed point. A particularly important application of the main result concerns families of hyperbolic saddles, where Siegel's theorem is too fragile, at least in the analytic category. By relaxing on the formal normal form we obtain analytic conjugacies. Since we consider families, it is more convenient to state some results for analytic maps on a Banach space; this gives no extra complications. As an example we treat a family passing through a 1 : -1 resonant saddle. (C) 2010 Elsevier Inc. All rights reserved.

We consider one parameter families of analytic vector fields and diffeomorphisms, including for a parameter value, say $\varepsilon = 0$, the product of rotations in $\R^{2m}\times \R^n$ such that for positive values of the parameter the origin is a hyperbolic point of saddle type. We address the question of determining the limit stable invariant manifold when $\varepsilon$ goes to zero as a subcenter invariant manifold when $\varepsilon = 0$.

[Bonckaert, Patrick; De Maesschalck, Peter] Hasselt Univ, B-3590 Diepenbeek, Belgium. [Caubergh, Magdalena] Univ Autonoma Barcelona, Bellaterra 08193, Spain.

We study normal forms of isolated singularities of vector fields in Rn or Cn. When all eigenvalues of the linear part of the vector field are nonzero, one can eliminate all so-called nonresonant terms from the equation provided some spectral condition (like Siegel) is satisfied. In this paper, we discuss the case where there is one zero eigenvalue (in that case Siegel's condition is not satisfied), and show that the formal normalizing transformations are either convergent or divergent of at most Gevrey type. In some cases, we show the summability of the normalizing transformations, which leads to the existence of analytic normal forms in complex sectors around the singularity.

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.

We give sufficient conditions on the spectrum at the equilibrium point such that a Gevrey-s family can be Gevrey-s conjugated to a simplified form, for 0 <= s <= 1. Local analytic results (i.e. s = 0) are obtained as a special case, including the classical Poincare theorems and the analytic stable and unstable manifold theorem. As another special case we show that certain center manifolds of analytic vector fields are of Gevrey-1 type. We finally study the asymptotic properties of the conjugacy on a polysector with opening angles smaller than s pi by considering a Borel-Laplace summation.

We consider one parameter families of vector fields depending on a parameter ɛ such that for ɛ=0 the system becomes a rotation of R2×Rn around {0}×Rn and such that for ɛ>0 the origin is a hyperbolic singular point of saddle type with, say, attraction in the rotation plane and expansion in the complementary space. We look for a local subcenter invariant manifold extending the stable manifolds to ɛ=0. Afterwards the analogous case for maps is considered. In contrast with the previous case the arithmetic properties of the angle of rotation play an important role.

We consider one-parameter families of maps close to a linear rotation in R2+n with conditions that imply that they axe weakly attracting in the rotation plane and weakly repelling in the 'rotation axis'. We get that the unstable manifolds converge to the 'rotation axis' and that, in the case of time one maps of vector fields, the stable manifolds converge to an independent of the parameter rotationally symmetric surface when the maps approach the rotation.

We study local analytic models for hyperbolic singularities of analytic families of real analytic vector fields. We emphasize on those cases where the normal form theorems of Poincaré and Siegel do not apply, for example for a family of saddles. We also consider the preservation of possible symmetries.

In this article, we develop some techniques to linearize families of smooth vector fields in a neighbourhood of a hyperbolic equilibrium point. In particular, we present the linearizing conjugacy in an explicit way and describe the smoothness of the conjugacy in terms of the eigenvalues of the vector fields.

In this paper we develop an explicit normal form conjugacy procedure, called an 'LMT normal form', to study linearization of a smooth vector field in the neighbourhood of a hyperbolic equilibrium point with resonant eigenvalues. We give an asymptotic expression for such a linearization in terms of functions of Logarithmic Mourtada type.