Bulletin of the Belgian Mathematical Society - Simon Stevin

The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web

Henk Broer, Carles Simó, and Renato Vitolo

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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.

Article information

Bull. Belg. Math. Soc. Simon Stevin, Volume 15, Number 5 (2008), 769-787.

First available in Project Euclid: 5 December 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34K18: Bifurcation theory 37D45: Strange attractors, chaotic dynamics 35B34: Resonances

Quasi-periodic bifurcations Invariant two-torus Ruelle-Takens scenario


Broer, Henk; Simó, Carles; Vitolo, Renato. The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web. Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 5, 769--787. doi:10.36045/bbms/1228486406. https://projecteuclid.org/euclid.bbms/1228486406

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