Centers and isochronous centers for two classes of generalized seventh and ninth systems
type: Item
instance of: scholarly article
title: Centers and isochronous centers for two classes of generalized seventh and ninth systems
zbMATH Open document ID: 1222.34032
author: Jaume Llibre
author: Claudia Valls
published in: Journal of Dynamics and Differential Equations
publication date: 2011-01-07T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/226095
review text: The authors study centers and isochronous centers for planar polynomial systems of odd degree \(d\geq 7\) that, in complex notation, can be written as \[ \dot z= (\lambda+i)z+(z\bar z)^{(d-7-2j)/2}\left(Az^{5+j}\bar z^{2+j} +B z^{4+j}\bar z^{3+j}+Cz^{3+j}\bar z^{4+j}+D\bar z^{7+2j}\right), \] where \(j\in\{0,1\}\), \(\lambda\) is a real parameter and \(A,B,C\) and \(D\) are complex parameters. Note that it is a family of systems with homogeneous non-linearities. Therefore, by using the Cherkas transformation, it can be transformed into an Abel equation. Both, the expression of the system in polar coordinates and the associated Abel equation, are used along the paper to study the centers and the isochronous centers. All centers found in the family are either Hamiltonian or reversible with respect to a straight line.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C27
zbMATH DE Number: 5833834
zbMATH Keywords: center
zbMATH Keywords: isochronous center
zbMATH Keywords: reversible center
zbMATH Keywords: Lyapunov quantities
zbMATH Keywords: Abel equation
MaRDI profile type: Publication
OpenAlex ID: W1992225889
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DOI: 10.1007/S10884-010-9175-0
description: scientific article; zbMATH DE number 5833834