Limit cycles for multidimensional vector fields. The elliptic case
type: Item
instance of: scholarly article
title: Limit cycles for multidimensional vector fields. The elliptic case
zbMATH Open document ID: 1038.34035
DOI: 10.1023/A:1023241823216
author: Marcin Bobieński
author: Henryk Żołądek
published in: Journal of Dynamical and Control Systems
publication date: 2003-06-09T00:00:00Z
review text: Consider the \((m+2)\)-dimensional system of real ordinary differential equations \[ \dot x=H_y+ \mathbf{z}\cdot\mathbf{R}+ \varepsilon P, \qquad \dot y=-H_x+\mathbf{z}\cdot \mathbf{S}+\varepsilon Q, \qquad \dot{ \mathbf z}=\mathbf{Az}+ \varepsilon\mathbf{b}, \] where \(\varepsilon\) is a small parameter, \(H=H(x,y)=y^2-x^3+3x\) is the Hamiltonian, \(P=P(x,y;\mathbf{z},\varepsilon)\) and \(Q=Q(x,y;\mathbf{z},\varepsilon)\) are scalar functions, \({\mathbf A}(x,y;\mathbf{z})\) is a matrix function, and \({ \mathbf R}={\mathbf R}(x,y;\mathbf{z}),\) \({\mathbf S}={\mathbf S}(x,y;\mathbf{z})\) and \({\mathbf b}={\mathbf b}(x,y;\mathbf{z},\varepsilon)\) are vector functions. All these functions are polynomial in \((x,y)\) and analytic in \((\mathbf{z},\varepsilon)\) near \(({\mathbf 0},0).\) Notice that when \(\varepsilon=0\) the above system has, in the plane \({\mathbf z}={\mathbf 0}\), a continuum of periodic orbits \(\gamma_h\) given by \(\{(x,y): H(x,y)=h,\, x<1 \},\) for \(-2<h<2.\) The authors study which of these periodic orbits remains for the perturbed system when \(\varepsilon\) is small enough. As a first step, the authors prove that, if Re\(| \mu_i(x,y)| >2\sqrt3,\,i=1\ldots,m,\) where \(\mu_i(x,y)\) are the eigenvalues of \({\mathbf A}(x,y;{\mathbf 0}),\) then the perturbed system has, for \(\varepsilon\) small enough, a \(C^1\) invariant surface \(L_\varepsilon\) given by an equation of the form \({\mathbf z}=\varepsilon {\mathbf g}(x,y)+ O(\varepsilon^2),\) where \({\mathbf g}\) satisfies the differential equation \(d{\mathbf g}/dt:={\mathbf g}_xH_y-{\mathbf g}_yH_x={\mathbf A g}+{\mathbf b},\) and all expressions are evaluated at \( {\mathbf z}={\mathbf 0}\). The key point for the proof of the above result is the notion of normal hyperbolicity developed by Hirsch, Pugh and Shub. Then, it is proved that the simple zeroes of the Poincaré-Pontryagin-Melnikov-Arnold integral defined for \(h\in(-2,2)\) as \[ I(h)=\int_{\gamma_h}(Q+{\mathbf g}\cdot {\mathbf S}…
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 1924863
zbMATH Keywords: Poincaré-Pontryagin-Melnikov-Arnold integral
zbMATH Keywords: limit cycle
zbMATH Keywords: normal hyperbolicity
MaRDI profile type: Publication
full work available at URL: a:1023241823216
OpenAlex ID: W84819111
description: scientific article; zbMATH DE number 1924863