Multidimensional formal Takens normal form

Stróżyna, Ewa; Żołądek, Henryk

doi:10.36045/bbms/1228486416

Cited by 4 publications

(6 citation statements)

References 6 publications

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“…Somewhat before that, we have the results of [2] on the formal structure of the normal forms with a nilpotent linear part, using representation theory of sl(2, C). More recently [11] and [8] have also made contributions to this subject in the multidimensional case, on the formal level.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

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Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Bonckaert¹,

Verstringe²

2012

Ann. inst. Fourier

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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-α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey-1 + α type with the use of a Gevrey-1 + α 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.

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Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…More recently [11] and [8] have also made contributions to this subject in the multidimensional case, on the formal level.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Bonckaert¹,

Verstringe²

2012

Ann. inst. Fourier

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“…In dimension n = 2, F. Takens [Tak73] has shown that the space of normal forms associated to S = y∂ x , C k , can be chosen to be the vector space generated by x k ∂ x and x k ∂ y , k ≥ 2. This has been generalized recently by E. Stróżyna and H.Żo ladek [SŻ08] to the case n ≥ 2.…”

Section: Formal Normal Formsmentioning

confidence: 83%

“…A more geometric proof was given in [Lor06]. It was shown also by E. Stróżyna and H.Żo ladek that the equivalent theorem in a higher dimensional setting is false [SŻ11,SŻ08].…”

Section: Analytic Normal Form Problemmentioning

confidence: 95%

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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“…Let us recall the result of [SZ2]. We consider germs in (C n , 0) of analytic vector fields of the form V = X + h.o.t.…”

Section: Introductionmentioning

confidence: 99%

Divergence of the reduction to the multidimensional nilpotent Takens normal form

Stróżyna

Żołądek

2011

Nonlinearity

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In Stróżyna and Żoła ¸dek (2008 Bull. Belgian Math. Soc. Simon Stevin 15 927-34) a generalization of the Takens normal form for a nilpotent singularity of a vector field in C n was obtained. Here we present an example where the corresponding normalizing series is divergent. This indicates that the generalized Takens normal form for a general nilpotent singularity in C n , n 3, is non-analytic.

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Multidimensional formal Takens normal form

Cited by 4 publications

References 6 publications

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

Divergence of the reduction to the multidimensional nilpotent Takens normal form

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