Divergence of a real normalizing transformation
type: Item
instance of: scholarly article
title: Divergence of a real normalizing transformation
zbMATH Open document ID: 0544.34006
DOI: 10.1007/BF01145469
published in: Mathematical Notes
publication date: 1982-01-01T00:00:00Z
review text: The following system is considered: (1) \(\dot z{}_ j=z_ j(\lambda_ j+f_ j(z_ 1,z_ 2))\), \(j=1,2\), where \(f_ j\) are analytic at zero \(z_ 1=0\), \(z_ 2=0\), \(f_ j(0,0)=0\) and \(\lambda_ 1=-\lambda_ 2\neq 0\). There exists a formal coordinate transformation (2) \(z_ j=w_ j(1+h_ j(w_ 1,w_ 2)), j=1,2\), reducing the system (1) to the normal form (3) \(\dot w_ j=w_ j(\lambda_ j+g_ j(w_ 1w_ 2)), j=1,2\), where \(h_ j\) and \(g_ j\) are formal power series of \(w_ 1,w_ 2\) without free members and \(g_ j\) depends only on the product \(w_ 1w_ 2\). Here only the case Re \(\lambda {}_ 1=Re \lambda_ 2\) is considered. The following theorem is proved: For each analytic normal form (3) satisfying some ''reality'' conditions, there exists an analytic system (1), formally equivalent to system (3) and for which any normalizing transformation (2) is divergent.
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 3865716
zbMATH Keywords: coordinate transformation
zbMATH Keywords: analytic normal form
author: A. D. Bryuno
MaRDI profile type: Publication
cites work: Normal form of differential equations with a small parameter
cites work: Q5679142
cites work: Researches on the restricted three-body problem
cites work: Q3852504
cites work: Q3891009
Recommended article: Q4283485
Recommended article: Q3035607
Recommended article: Divergence for the normalization of real analytic glancinghypersurfaces
Recommended article: Divergence points and normal numbers
Recommended article: Q3797537
Recommended article: On the divergence of Birkhoff normal forms
Recommended article: Norm of the Cauchy transform
Recommended article: Q3760903
Recommended article: On hyperbolic transformations to normality
description: scientific article; zbMATH DE number 3865716