Formal Normal Form Research Articles

Complete transversal and formal normal forms of germs of vector fields

In this work, inspired by the technique of the complete transversal, used for the classification of plane branches, developed by Hefez, A. and Hernandes, M., as well as Bruce, J.W., Kirk, N.P. and du Plesis, A.A., study the singularities of applications, we establish a classification of vector fields through their normal forms. In the case of vector fields with non zero linear part in $(\mathbb{C}^{2}, 0) $ and nilpotent fields in $(\mathbb {C}^{n}, 0), n\geq 2$ we recover the classical normal forms for those fields, and we provide a formal normal form different from Takens in dimension 2. Likewise, we obtain the normal form for the vector fields in $(\mathbb{C},0)$ of any multiplicity.

Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions

This article is concerned with the geometry of germs of real analytic surfaces in $$({\mathbb {C}}^2,0)$$ having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in $$({\mathbb {C}}^2,0)$$ having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $$\{\tau _1,\tau _2\}$$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $$\{z_1z_2=const\}$$ (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity * *Supported by the Polish NCN OPUS Grant 2017/25/B/ST1/00931

In Stróżyna E and Żołądek H (2015 Moscow Math. J. 15 141) a complete formal normal forms for germs of two-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle-nodes) the normal form is nonanalytic due to bad properties of some homological operators associated with the first nontrivial term in the orbital normal form.

О формальной нормальной форме ростков полугиперболических отображений на плоскости

There are consider the problem of constructing an analytical classification holomorphic resonance maps germs of Siegel-type in dimension 2. Namely, semi-hyperbolic maps of general form: such maps have one parabolic multiplier (equal to one), and the other hyperbolic (not equal in modulus to zero or one). In this paper, the first stage of constructing an analytical classification by the method of functional invariants is carried out: a theorem on the reducibility of a germ to its formal normal form by "semiformal" changes of coordinates is proved. The one-time shift along the saddle node vector field (the formal normal form in the problem of the analytical classification of saddle-node vector fields on a plane) is chosen as the formal normal form.

Equivariant decomposition of polynomial vector fields

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.

Convergence of the Chern–Moser–Beloshapka normal forms

Abstract In this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of ℂ N {\mathbb{C}^{N}} of codimension d ≥ 1 {d\geq 1} under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case d = 1 {d=1} our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

Real submanifolds of maximum complex tangent space at a CR singular point, II

We study a germ of real analytic n-dimensional submanifold of $C^n$ that has a complex tangent space of maximal dimension at a CR singularity. Under the condition that its complexification admits the maximum number of deck transformations, we first classify holomorphically its quadratic CR singularity. We then study its transformation to a normal form under the action of local (possibly formal) biholomorphisms at the singularity. We first conjugate formally its associated reversible map $\sigma$ to suitable normal forms and show that all these normal forms can be divergent. We then construct a unique formal normal form under a non degeneracy condition.

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed.

Normalization in Lie algebras via mould calculus and applications

We establish Ecalle's mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincar{\'e}-Dulac formal normal forms for a vector field around an equilibrium point, a formal infinite-order multiphase averaging procedure for vector fields with fast angular variables (Hamiltonian or not), or the construction of Birkhoff normal forms both in classical and quantum situations. As a by-product we obtain, in the case of harmonic oscillators, the convergence of the quantum Birkhoff form to the classical one, without any Diophantine hypothesis on the frequencies of the unperturbed Hamiltonians.

Intrinsic character of Stokes matrices

Two germs of linear analytic differential systems xk+1Y′=A(x)Y with a non-resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in S allows supposing that R+ lies in the intersection of two sectors. Reordering of the coordinates of Y allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in x is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.

Normal forms and embeddings for power-log transseries

Dulac series are asymptotic expansions of first return maps in a neighborhood of a hyperbolic polycycle. In this article, we consider two algebras of power-log transseries (generalized series) which extend the algebra of Dulac series. We give a formal normal form and prove a formal embedding theorem for transseries in these algebras.

Mapping properties of normal forms transformations for water waves

It is a well known fact, dating from Zakharov (J Appl Mech Tech Phys 9:190–194, 1968), that the equations for free surface water waves can be posed as a Hamiltonian system. It is also known that the Hamiltonian involves no three-wave resonances, and that in the case of dimension \(n=2\) and in infinite depth, the four-wave resonances have a special integrable structure (Dyachenko and Zakharov, Phys Lett A 190:144–148, 1994; Craig and Worfolk, Phys D 84:513–531, 1995). The implication, at least on a formal level, is that cubic terms and nonresonant quartic terms can be removed from the Hamiltonian by suitable canonical transformations, and thus these terms do not enter the equations of motion. The resulting Hamiltonian is said to be in third order (respectively, fourth order) Birkhoff normal form. In this article we study the function space mapping properties of the third and fourth order canonical transformations to Birkhoff normal form, in the case of spatially periodic data in dimension \(n=2\). We find that the third order normal forms transformation is defined in a neighborhood of zero in the Hilbert space \(H^r\times H^r\) for any \(r > 3/2\), and somewhat surprisingly the transformation is based on the time-one flow of Burger’s equation. For the fourth order Birkhoff normal form, we derive a clean expression for the formal normal form, but also we find that the vector field of the Hamiltonian flow that would give rise to this transformation is not bounded on any Sobolev space of the form \(H^r\times H^s\). Alternatively, we give a partial normal form, which results in the elimination of many, but not all, of the nonresonant quartic terms of the Hamiltonian.

Normal form of holomorphic vector fields with an invariant torus under Brjuno’s A condition

We consider the holomorphic normalization problem for a holomorphic vector field in the neighborhood of the product of a fixed point and an invariant torus. Supposing that the vector field is a perturbation of a linear part around the fixed point and of a rotation on the invariant torus (the unperturbed vector field is called the quasi-linear part of the perturbed one), it was shown by J.Aurouet that the system is holomorphically linearizable if there are no exact resonances in the quasi-linear part and if the quasi-linear part satisfies to Brjuno's arithmetical condition. In the presence of exact resonances, a conjecture by Brjuno states that the system will still be holomorphically conjugated to a normal form under the same arithmetical condition and a strong algebraic condition on the formal normal form. This article proves this conjecture.

Analysis of a slow–fast system near a cusp singularity

This paper studies a slow–fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results.

Formal normal form of [formula omitted] slow–fast systems

An Ak slow–fast system is a particular type of singularly perturbed ODE. The corresponding slow manifold is defined by the critical points of a universal unfolding of an Ak singularity. In this note we propose a formal normal form of Ak slow–fast systems.

Normal forms for CR singular codimension-two Levi-flat submanifolds

Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to ${\mathbb{C}}^m$, $m > 2$, of Bishop surfaces in ${\mathbb{C}}^2$. Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic. We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater. In fact, the nondegenerate submanifolds, i.e. higher order purturbations of $z_m=\bar{z}_1z_2+\bar{z}_1^2$, have no analogue in dimension 2. We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case. Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group. In general, we find a formal normal form in ${\mathbb{C}}^3$ in the nondegenerate case that shows infinitely many formal invariants.

Analytic Integrability of Some Examples of Degenerate Planar Vector Fields

This paper is devoted to the classification of analytic integrable cases of two families of degenerate planar vector fields with a monodromic singular point at the origin. This study falls in the still open degenerate center problem. This classification can be done using the formal normal form theory and knowing a suitable normal form of any differential systems associated to each family.

Big denominators and analytic normal forms

Abstract We study the regular action of an analytic pseudo-group of transformations on the space of germs of various analytic objects of local analysis and local differential geometry. We fix a homogeneous object F 0 and we are interested in an analytic normal form for the whole affine space { F 0 + h.o.t. } ${\lbrace F_0 + \text{h.o.t.}\rbrace }$ . We prove that if the cohomological operator defined by F 0 has the big denominators property and if a formal normal form is well chosen, then this formal normal form holds in analytic category. We also define big denominators in systems of nonlinear PDEs and prove a theorem on local analytic solvability of systems of nonlinear PDEs with big denominators. Moreover, we prove that if the denominators grow “relatively fast”, but not fast enough to satisfy the big denominators property, then we have a normal form, respectively local solvability of PDEs, in a formal Gevrey category. We illustrate our theorems by explanation of known results and by new results in the problems of local classification of singularities of vector fields, non-isolated singularities of functions, tuples of germs of vector fields, local Riemannian metrics and conformal structures.

A normal form for a real 2-codimensional submanifold in [formula omitted] near a CR singularity

We construct a formal normal form for a class of real submanifolds M⊂CN+1 of codimension 2 near a CR singularity approximating the sphere. Our result gives a generalization of Huang–Yin’s normal form in C2 to the higher dimensional analog case.

Dynamics of one-resonant biholomorphisms

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in \mathbb C^n whose differentials have one-dimensional family of resonances in the first m eigenvalues, m\leq n (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.