Global stability of saddle-node bifurcation of a periodic orbit for vector fields
type: Item
instance of: scholarly article
title: Global stability of saddle-node bifurcation of a periodic orbit for vector fields
zbMATH Open document ID: 0831.58036
author: Sergio Plaza S.
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1995-09-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73342
review text: In a bifurcation value, the global stability of families of \(C^\infty\) vector fields, which have a generically unfolding saddle-node periodic orbit is studied. Let \(M\) be a \(C^\infty\) closed manifold, \({\mathcal X}^\infty (M)\) be the space of \(C^\infty\) vector fields on \(M\) and \({\mathcal X}_1^\infty (M)\) be the space of \(C^\infty\) arcs \(\xi: I=[-1, 1]\to {\mathcal X}^\infty (M)\) of vector fields, both endowed with the \(C^\infty\) Whitney topology. Assuming that the arc \(\xi= \{X_\mu \}\), \(\mu\in I\), satisfies the properties: (1) for each \(\mu\in I\), the non-wandering set of \(X_\mu\) consists of a finite number of critical elements (i.e. singularities and periodic orbits) of \(X_\mu\), (2) there are no cycles among the critical elements of \(X_\mu\), (3) there is \(\overline {\mu}\in I\) such that \(X_{\overline {\mu}}\) has a saddle-node periodic orbit, under certain additional conditions, the author obtains results on the global stability, non-stability of \(\{X_\mu\}\) at a bifurcation value \(\overline {\mu}\) and a modulus of stability for \(X_{\overline {\mu}}\). The results of the paper generalize some results of \textit{I. Malta} and \textit{J. Palis} [Lect. Notes Math. 898, 212-229 (1981; Zbl 0482.58023)]\ obtained for the case \(\dim M=2\).
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 778924
zbMATH Keywords: saddle-node periodic orbit
zbMATH Keywords: one-parameter families of vector fields
zbMATH Keywords: bifurcations
zbMATH Keywords: stability
zbMATH Keywords: modulus of stability
reviewed by: Eugene Ershov
MaRDI profile type: Publication
OpenAlex ID: W2316662119
cites work: Q3940526
cites work: Q3889814
cites work: Invariant manifolds
cites work: Q3769660
cites work: Stability of Morse-Smale vector fields on manifolds with boundary
cites work: Q3940527
cites work: Spiralling invariant manifolds
cites work: Bifurcations and stability of families of diffeomorphisms
cites work: Stability of parametrized families of gradient vector fields
cites work: Q3927932
cites work: Global phenomena in bifurcations of dynamical systems with simple recurrence
cites work: Normal forms for certain singularities of vectorfields
DOI: 10.5802/AFST.787
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description: scientific article; zbMATH DE number 778924