Normal forms for periodic orbits of real vector fields
type: Item
instance of: scholarly article
title: Normal forms for periodic orbits of real vector fields
zbMATH Open document ID: 1157.34032
author: Ping Wang
author: Hao Wu
author: Weigu Li
publication date: 2008-09-09T00:00:00Z
review text: The authors consider the differential system \[ \dot{x}=A\left( t\right) x+f\left( x,t\right)\;,\;\dot{t}=1+g\left( x,t\right), \] where \(A\left( t\right)\in C^{\infty}\left( T_{2\pi},\mathbb{R}^{n\times n}\right)\), \(f\left( x,t\right)\in C^{\infty}\left( U_r\times T_{2\pi},\mathbb{R}^n\right)\) and \(g\left( x,t\right)\in C^{\infty}\left( U_r\times T_{2\pi},\mathbb{R}\right)\) which satisfy \(f\left( x,t\right) = O\left( \| x\| ^2\right)\) and \(g\left( x,t\right) = O\left(\| x\| \right)\) as \(x\to 0\). Note, \(U_r :=\{ x:\| x\| <r\}\subset\mathbb{R}^n\) and \(T_{2\pi}\) is the circle of length \(2\pi\). The main theorem of the paper reads: If the Floquet exponents of the above system are hyperbolic and satisfy non-resonance conditions, then the system is \(C^{\infty}\) locally equivalent to the differential system \[ \dot{y}=A\left( s\right) y+\tilde{f}\left( y,s\right),\quad \dot{s}=1. \]
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 34C41
zbMATH DE Number: 5323422
zbMATH Keywords: normal form
zbMATH Keywords: homotopy method
zbMATH Keywords: periodic orbit
zbMATH Keywords: foliation
reviewed by: Andrzej Piatkowski
MaRDI profile type: Publication
full work available at URL: s10114-007-0982-0
OpenAlex ID: W1995079079
cites work: Integral manifolds of perturbed differential systems
cites work: Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows.
cites work: Q4210478
cites work: Finitely-smooth normal forms of local families of diffeomorphisms and vector fields
cites work: Q3997066
cites work: Normally hyperbolic invariant manifolds in dynamical systems
DOI: 10.1007/S10114-007-0982-0
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5323422