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December 2008 The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web
Henk Broer, Carles Simó, Renato Vitolo
Bull. Belg. Math. Soc. Simon Stevin 15(5): 769-787 (December 2008). DOI: 10.36045/bbms/1228486406

Abstract

A model map $Q$ for the Hopf-saddle-node (HSN) bifurcation of fixed points of diffeomorphisms is studied. The model is constructed to describe the dynamics inside an attracting invariant two-torus which occurs due to the presence of quasi-periodic Hopf bifurcations of an invariant circle, emanating from the central HSN bifurcation. Resonances of the dynamics inside the two-torus attractor yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps, where a web of hyperbolic periodic points is expected to occur inside the two-torus attractor. It is conjectured that heteroclinic intersections of the invariant manifolds of the saddle periodic points may give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. To understand this phenomenon, a simple model map of the standard two-torus is developed and studied and the relations with the starting model map $Q$ are discussed.

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Henk Broer. Carles Simó. Renato Vitolo. "The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web." Bull. Belg. Math. Soc. Simon Stevin 15 (5) 769 - 787, December 2008. https://doi.org/10.36045/bbms/1228486406

Information

Published: December 2008
First available in Project Euclid: 5 December 2008

zbMATH: 1154.37319
MathSciNet: MR2484131
Digital Object Identifier: 10.36045/bbms/1228486406

Subjects:
Primary: 34K18, 35B34, 37D45

Rights: Copyright © 2008 The Belgian Mathematical Society

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Vol.15 • No. 5 • December 2008
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