Divergence and summability of normal forms of systems of differential equations with nilpotent linear part

Canalis-Durand, Mireille; Schäfke, Reinhard

doi:10.5802/afst.1079

Cited by 9 publications

(8 citation statements)

References 6 publications

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“…The corresponding orbital normal forms, i.e., the two choices of Φ, agree with those used in [9]. Remark 5.…”

Section: The Normal Formsupporting

confidence: 62%

“…In the case of the Bogdanov-Takens singularity (B-T singularity), i.e., with nonzero nilpotent linear part, the classical Takens prenormal form [22] (y + a(x)) ∂ ∂x + b(x) ∂ ∂y is analytic (see [14,17]). But the formal orbital normal form in the cusp case y ∂ ∂x + 3 2 x 2 ∂ ∂y + • • • turns out non-analytic (see [9] and section 7 below).…”

Section: Introductionmentioning

confidence: 99%

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

Stróżyna

Żołądek

2021

Nonlinearity

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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.

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“…The corresponding orbital normal forms, i.e., the two choices of Φ, agree with those used in [9]. Remark 5.…”

Section: The Normal Formsupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

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

Stróżyna

Żołądek

2021

Nonlinearity

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“…on dit que F est une singularité nilpotente ; c'est le cas de l'exemple en haut à droite dans la figure 5.3. Ces singularités ont fait l'objet de nombreux travaux (voir [147,58,74,123,116,144,14,212,211,213,162,39,145,81,141]) qui permettent aujourd'hui de bien les comprendre. Leur classification analytique se ramène cependant, dans le cas le plus simple, au problème ouvert discuté dans la section 2.6.…”

Section: Bestaire Préliminaire Des Singularités Non Dégénéréesunclassified

Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux

Loray¹

2021

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Classification formelle et propriétés algébriquesNotons Diff(C, 0) le groupe pour la composition des "germes de difféomorphismes formels" fixant 0 ∈ C :Nous décrivons les propriétés algébriques de Diff(C, 0), ou plus généralement de Diff(C, 0). Nous donnons notamment une classification des éléments et sous-groupes résolubles de Diff(C, 0) à conjugaison près dans Diff(C, 0) (classification formelle). Nous terminerons par une description assez sommaire des sous-groupes non résolubles et aborderons quelques questions ouvertes dans ce domaine.2.1 Sous-algèbres de Lie de X (C, 0)On note X (C, 0) l'algèbre de Lie des germes de champs de vecteurs holomorphes en 0 ∈ C :Une sous-algèbre L de X (C, 0) est un sous-espace vectoriel complexe stable par crochet de Lie. On dit que L est transitive si l'un au moins de ses éléments ne s'annule pas en 0 ; on dira qu'elle est intransitive sinon.Commençons par rappeler le résultat classique de Sophus Lie sur la classification des géométries de la droite (voir [109])Remarque 2.4. Dans la suite, on notera X p,λ la forme normale obtenue dans le cas dégénéré(2.1)Notons que (z p+1 − λ 2iπ z 2p+1 )∂ z est conjugué à X p,λ (voir la démonstration précédente) et aurait tout aussi bien pu convenir comme forme normale. Cependant, X p,λ a l'avantage que sa forme duale ω p,λ = dz z p+1 + λ 2iπ dz z admet une primitive bien plus simple, à savoir :Cette transformation (multiforme) redresse X p,λ sur le champ constant ∂ z et sera maintes fois utilisée plus loin.

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“…If V = N + R ≥2 , a germ of holomorphic vector field at the origin of C n , can be embedded into a formal sl(2, C)-triple, that is if there are formal vector There are other results concerning the problem of classification of perturbations not of nilpotent vector fields but rather of quasihomogenous vector fields with a nilpotent linear part at the origin. They are all in dimension 2 (see for instance [MC88,CDS04]) and are not immediately concerned with holomorphic conjugacy (in a neighborhood) at the origin to a normal form.…”

Section: Resultsmentioning

confidence: 99%

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Stolovitch¹,

Verstringe

2016

Regul. Chaot. Dyn.

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We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension n ≥ 3. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Stróżyna-Żo ladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and sl 2 (C)-representations.

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Divergence and summability of normal forms of systems of differential equations with nilpotent linear part

Cited by 9 publications

References 6 publications

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

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

Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

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Divergence and summability of normal forms of systems of differential equations with nilpotent linear part

Cited by 9 publications

References 6 publications

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

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

Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

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Product

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

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