Dulac time of a resonant saddle in the loud family
type: Item
instance of: scholarly article
title: Dulac time of a resonant saddle in the loud family
zbMATH Open document ID: 1455.34040
author: M. Saavedra
published in: Journal of Differential Equations
publication date: 2020-07-02T00:00:00Z
review text: For the normalized Loud family of reversible systems \[\begin{aligned} \dot{x} &=-y+xy \\ \dot{y} &=x+Dx^2+Fy^2, \end{aligned}\] the author justifies the behavior of the period function on the outer boundary of the period annulus which surrounds the center at the origin. Namely, taking \(\mu=(D,F)\in W:=(-1,-\frac12)\times (\frac13,\frac23)\), the first few terms in the expansion of the related period function \(T(\mu,s)\) are calculated. Then using them and uniformity with respect to \(\mu\), it is proved that \(T(\mu,s)\) is regular and moreover \(\partial_sT<0\) in a neighborhood of the segment \(S:=W\cap\{F=\frac12\}\). In such a way the role of \(S\) in the bifurcation diagram (as given initially by \textit{P. Mardešić} et al. [J. Differ. Equations 224, No. 1, 120--171 (2006; Zbl 1092.34020)] is clarified.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7216735
zbMATH Keywords: period function
zbMATH Keywords: resonant saddle
zbMATH Keywords: asymptotic expansions
reviewed by: Iliya D. Iliev
MaRDI profile type: Publication
full work available at URL: j.jde.2020.05.023
OpenAlex ID: W3035580887
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DOI: 10.1016/J.JDE.2020.05.023
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description: scientific article; zbMATH DE number 7216735