Symmetries and Convergence of Normalizing Transformations

Bruno, A D; Walcher, Sebastian

doi:10.1006/jmaa.1994.1163

Cited by 52 publications

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“…When n = 2 the above theorem has been obtained by Bruno and Walcher [4]. Their proof is based on Bruno's results [3], and therefore uses the fast convergence method directly or indirectly, in contrast to our method.…”

Section: Introductionmentioning

confidence: 90%

“…In the geometrical approach, one replaces Diophantine conditions by symmetries and first integrals, and replaces the fast convergence method by arguments of geometrical nature. Some recent papers on the subject like [1,4,5,9,10] contain some geometrical ingredients (commuting vector fields, symmetry groups), but the methods used remain mostly analytical. In the present paper, we will follow the geometrical approach in a more substantial way.…”

Section: Introductionmentioning

confidence: 99%

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Convergence versus integrability in Poincaré-Dulac normal form

Zung

2002

Mathematical Research Letters

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Abstract. We show that, to find a Poincaré-Dulac normalization for a vector field is the same as to find and linearize a torus action which preserves the vector field. Using this toric characterization and other geometrical arguments, we prove that any local analytic vector field which is integrable in the nonHamiltonian sense admits a local convergent Poincaré-Dulac normalization. These results generalize the main results of our previous paper [12] from the Hamiltonian case to the non-Hamiltonian case. Similar results are presented for the case of isochore vector fields.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Convergence versus integrability in Poincaré-Dulac normal form

Zung

2002

Mathematical Research Letters

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“…In recent years, it has turned out that the question of convergence is closely related to the question of the existence of nontrivial "infinitesimal symmetries" for the given vector field. In fact, for two-dimensional systems it follows from results of Markhashov [11] and Bruno and Walcher [4] that there is a convergent transformation to normal form if and only if there is a nontrivial infinitesimal symmetry. Cicogna [5,6] extended parts of this result to higher dimension.…”

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confidence: 92%

On Convergent Normal Form Transformations in Presence of Symmetries

Walcher¹

2000

Journal of Mathematical Analysis and Applications

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We prove a convergence criterion for transformations to Poincaré-Dulac normal form that involves the centralizer of the given vector field. © 2000 Academic Press Since they were introduced by Poincaré and Dulac (and by Birkhoff for Hamiltonian systems), normal forms have proven to be a valuable tool in the local theory of ordinary differential equations. In many cases it turns out that a "partial" normal form (up to some finite degree in the Taylor expansion) is sufficient to provide an understanding of the structure of the solutions near a stationary point. Some problems, however, for instance the local analytic classification of vector fields, lead to the question whether a convergent transformation to normal form exists for a given vector field. It is known that often such a convergent transformation does not exist, and results by Bruno [2] indicate that convergence is indeed a rare phenomenon as soon as the normal form is not just equal to the linear part. (We will use this paper and the book [3] by Bruno as a basic reference.) In recent years, it has turned out that the question of convergence is closely related to the question of the existence of nontrivial "infinitesimal symmetries" for the given vector field. In fact, for two-dimensional systems it follows from results of Markhashov [11] and Bruno and Walcher [4] that there is a convergent transformation to normal form if and only if there is a nontrivial infinitesimal symmetry. Cicogna [5,6] extended parts of this result to higher dimension. Ito [9,10] proved convergence in the case of certain Hamiltonian systems under the assumption that there are sufficiently many integrals (in other words, sufficiently many canonical infinitesimal symmetries). For a survey of these and other results, see the paper [7] by Cicogna and Gaeta.

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“…Normal forms have been used to treat a range of dynamics problems, see for example Hsu (1983), Fredriksson & Nordmark (2000), Pelinovsky & Yang (2002), Leung & Zhang (2003) and Wang & Bajaj (2009). Mathematical aspects of the normal form technique have been studied by many authors, including Zhuravlev (1983), Walcher (1993), Bruno & Walcher (1994), Mayer et al (2004), Stolovitch (2009), Meyer et al (2009) and references therein.…”

Section: Introductionmentioning

confidence: 99%

Applying the method of normal forms to second-order nonlinear vibration problems

2010

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Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first-and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

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Symmetries and Convergence of Normalizing Transformations

Cited by 52 publications

References 0 publications

Convergence versus integrability in Poincaré-Dulac normal form

Convergence versus integrability in Poincaré-Dulac normal form

On Convergent Normal Form Transformations in Presence of Symmetries

Applying the method of normal forms to second-order nonlinear vibration problems

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