Local integrability of Poincaré-Dulac normal forms
type: Item
instance of: scholarly article
title: Local integrability of Poincaré-Dulac normal forms
zbMATH Open document ID: 1416.37053
DOI: 10.1134/S1560354718070080
author: Shogo Yamanaka
published in: Regular and Chaotic Dynamics
publication date: 2019-04-26T00:00:00Z
review text: This paper studies the analytic integrability of Poincaré-Dulac normal forms. The author considers dynamical systems of the form $\dot{x} = f(x)$ for $x\in\mathbb{C}^n$, where $n\ge 1$, $f:\mathbb{C}^n\rightarrow\mathbb{C}^n$ and $f(0) = 0$. Throughout it is assumed that $f$ has a power series expansion of the form $f(x)=\sum_{n=1}^{\infty} f^{i}$, where $f^i$ is a homogeneous vector field of degree $i$ for $i=1, 2,\dots$. If $A = Df(0)$, the matrix $A$ can be decomposed as $A = S + N$, where $S$ is semi-simple and $N$ is nilpotent. This leads to a decomposition $f^1=f^s(x)+f^n(x)$ where $f^s(x) = Sx$ and $f^n(x) = Nx$. The vector field \(f\) is said to be in Poincaré-Dulac normal form if the Lie bracket of $f^s$ and $f$, written $Df(x) f^s(x) - Df^s(x)f(x)$, vanishes for every $x\in\mathbb{C}^n$. \par The author reviews previous results for analytic integrability of the Poincaré-Dulac and Birkhoff normal forms [\textit{A. D. Bryuno}, Tr. Mosk. Mat. O.-va 25, 119--262 (1971; Zbl 0263.34003); Trans. Mosc. Math. Soc. 26, 199--239 (1974; Zbl 0283.34013); \textit{O. I. Bogoyavlenskij}, Commun. Math. Phys. 196, No. 1, 19--51 (1998; Zbl 0931.37028); \textit{Nguyen Tien Zung}, Math. Res. Lett. 9, No. 2--3, 217--228 (2002; Zbl 1019.34084); Ann. Math. (2) 161, No. 1, 141--156 (2005; Zbl 1076.37045)], and notes that there is a general transformation that produces a normal form for the original dynamical system but that transformation is not generally analytic. \par The main result of the paper is a sufficient condition for a dynamical system (having a form as described above) in Poincaré-Dulac normal form to be analytically integrable, and that condition is also necessary if certain technical criteria are met. The integrability conditions depend on the resonance degrees of the dynamical system. The author shows that systems with resonance degrees of 0 or 1 are integrable, but may not be if the resonance degrees are greater than 1. \par The author applies his theoretical resul…
Mathematics Subject Classification ID: 37J30
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37J20
zbMATH DE Number: 7048259
zbMATH Keywords: Poincaré-Dulac normal form
zbMATH Keywords: integrability
zbMATH Keywords: resonances
zbMATH Keywords: dynamical system
reviewed by: William J. Satzer Jr.
MaRDI profile type: Publication
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full work available at URL: s1560354718070080
OpenAlex ID: W2913846857
description: scientific article; zbMATH DE number 7048259