Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Stolovitch, Laurent; Verstringe, Freek

doi:10.1134/s1560354716040031

Cited by 4 publications

(3 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, these formulas may prove useful in the study of the convergence of the (first level) normal form where one needs to estimate the transformation through the Homological Equation. We refer the interested reader to [15], where parts of the proof show a striking similarity to the techniques employed in the present paper and we hope to improve on these results using the techniques described in [11].…”

Section: Discussionmentioning

confidence: 68%

Equivariant decomposition of polynomial vector fields

Mokhtari¹,

Sanders²

2019

Preprint

View full text Add to dashboard Cite

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 linear nilpotent. 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 present 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 case.

show abstract

Section: Discussionmentioning

confidence: 68%

Equivariant decomposition of polynomial vector fields

Mokhtari¹,

Sanders²

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…P Bonckaert and F Verstringe [8] proved that the formal normal form is Gervrey-1. In one special case, when the formal normal form is completely integrable, L Stolovich and F Verstringe [16] proved its analyticity. The latter two works are based on estimates, with use of techniques from [7,13].…”

Section: Introductionmentioning

confidence: 99%

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity ^*

Stróżyna

Żołądek

2021

Nonlinearity

View full text Add to dashboard Cite

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.

show abstract

“…Analytic normalization of an analytic vector field has close relations with complete integrability of the vector field; e.g., see [34,35,42,46]. We recall that two first integrals for v in B are called functionally independent when their gradients have a rank of 2 for almost everywhere; e.g., see [42, page 3553] and [34].…”

Section: Introductionmentioning

confidence: 99%

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Gazor

Mokhtari

Sanders

2019

Journal of Differential Equations

View full text Add to dashboard Cite

We introduce a sl 2 -invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations; one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.

show abstract

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Cited by 4 publications

References 43 publications

Equivariant decomposition of polynomial vector fields

Equivariant decomposition of polynomial vector fields

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity ^*

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Contact Info

Product

Resources

About

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Cited by 4 publications

References 43 publications

Equivariant decomposition of polynomial vector fields

Equivariant decomposition of polynomial vector fields

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity *

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Contact Info

Product

Resources

About

Analytic properties of the complete formal normal form for the Bogdanov–Takens singularity ^*