Limit cycles bifurcating from isochronous surfaces of revolution in \(\mathbb R^3\)
type: Item
instance of: scholarly article
title: Limit cycles bifurcating from isochronous surfaces of revolution in \(\mathbb R^3\)
zbMATH Open document ID: 1225.34039
author: Jaume Llibre
author: Salomón Rebollo-Perdomo
author: Joan Torregrosa
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-06-06T00:00:00Z
review text: Consider a differential system of the form \[ \dot x= X_0 (x) + \varepsilon X(x), \tag{1} \] where \(x \in \mathbb{R}^3\), \(X_0\), \(X: \mathbb{R}^3 \to \mathbb{R}^3\) are vector fields and \(\varepsilon\) is a real small parameter, and the dot denotes the derivative with respect to time. If we suppose that, for \(\varepsilon=0\), (1) has an isochronous invariant surface \(S\), that is, \(S\) is foliated by periodic orbits with the same period, then the following natural questions arise: For \(\epsilon \neq 0\) sufficiently small, does the differential system (1) posses limit cycles emerging form the periodic orbits of \(S\)? How to compute them and what about their number? A tool for treating these problems is the averaging theory. However, for a system of form (1) it is not possible to apply the classical averaging theory. In this paper, a result that goes back to \textit{I. G. Malkin} [Some problems of the theory of nonlinear oscillations (Russian). Moskau: Staatsverlag für technisch-theoretische Literatur (1956; Zbl 0070.08703)] and \textit{M. Roseau} [Vibrations non linéaires et théorie de la stabilité. Berlin-Heidelberg-New York: Springer-Verlag (1966; Zbl 0135.30603)] is applied and adapted to the problems that are studied in the paper. The authors study the number of limit cycles bifurcating from isochronous surfaces of revolution in \(\mathbb{R}^3\), when polynomial perturbations of arbitrary degree are considered.
reviewed by: Jaume Giné
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 5904480
zbMATH Keywords: Limit cycle
zbMATH Keywords: periodic orbit
zbMATH Keywords: isochronous set
zbMATH Keywords: averaging method
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.04.009
OpenAlex ID: W1976271680
cites work: Periodic solutions of nonlinear periodic differential systems with a small parameter
cites work: Limit cycles of differential equations
cites work: Limit cycles bifurcating from a two-dimensional isochronous cylinder
cites work: Limit cycles bifurcating from a \(k\)-dimensional isochronous center contained in \(\mathbb R^n\) with \(k\leqslant n\)
cites work: Limit Cycles Bifurcating from a 2-Dimensional Isochronous Torus in ℝ3
cites work: Q3785328
cites work: Q4001531
cites work: Averaging methods in nonlinear dynamical systems
cites work: Tchebycheff systems and best partial bases
cites work: Q4714154
DOI: 10.1016/J.JMAA.2011.04.009
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description: scientific article; zbMATH DE number 5904480