Perturbed normalizers and Melnikov functions
type: Item
instance of: scholarly article
title: Perturbed normalizers and Melnikov functions
zbMATH Open document ID: 1397.34058
author: Adriana Buică
published in: Journal of Mathematical Analysis and Applications
publication date: 2018-05-09T00:00:00Z
review text: Consider a \(C^{\infty}\) smooth family of real planar vector fields \(\left(X_\varepsilon\right)_\varepsilon\) of the form \[ X_\varepsilon (x) \, = \, X_0(x) + \varepsilon X_1(x) + \varepsilon^2 X_2(x) + \, \ldots \, + \varepsilon^n X_n(x) + \mathcal{O}\left(\varepsilon^{n+1}\right), \] for some integer \(n \geq 1\), \(x \in \mathbb{R}^2\) and \(|\varepsilon|\ll 1\). It is assumed that \(X_0(x)\) has a period annulus in a region \(\mathcal{P} \subset \mathbb{R}^2\). The number and location of the periodic orbits of \(X_0\) which ``persist'' in the family \(\left(X_\varepsilon\right)_\varepsilon\) as limit cycles is one of the questions to which more articles have been devoted. A classical method to study this bifurcation problem is the Melnikov functions. Given \(m\), an integer with \(m \geq 1\), the author provides a characterization of the property that the first \((m-1)\) Melnikov functions of \(\left(X_\varepsilon\right)_\varepsilon\) vanish identically. This characterization is given in terms of the existence of a smooth family of planar vector fields \(\left(U_\varepsilon\right)_\varepsilon\), called perturbed normalizers of order \(m\). A \textit{perturbed normalizer of order \(m\)} for the family \(\left(X_\varepsilon\right)_\varepsilon\) is a family of smooth vector fields \(\left(U_\varepsilon\right)_\varepsilon\) such that \(U_0\) is transversal to \(X_0\) in \(\mathcal{P}\) and \[ X_\varepsilon \wedge [X_\varepsilon,U_\varepsilon] \, = \, \mathcal{O}(\varepsilon^m) \, \quad \text{in \(\mathcal{P}\) for \(|\varepsilon|\ll 1\).} \] An effective formula to compute these vector fields \(\left(U_\varepsilon\right)_\varepsilon\) is provided, together with a formula for the derivative of the \(m\)-th order Melnikov function.
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 6867345
zbMATH Keywords: Melnikov functions
zbMATH Keywords: perturbed normalizers
zbMATH Keywords: planar vector field
zbMATH Keywords: Lie bracket
MaRDI profile type: Publication
full work available at URL: j.jmaa.2018.04.004
OpenAlex ID: W2795993454
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Wikidata QID: Q130041517
DOI: 10.1016/J.JMAA.2018.04.004
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description: scientific article; zbMATH DE number 6867345