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October  2010, 14(3): 871-905. doi: 10.3934/dcdsb.2010.14.871

Chaos and quasi-periodicity in diffeomorphisms of the solid torus

Henk W. Broer 1, , Carles Simó 2, and  Renato Vitolo 3,

1. 

Dept. of Mathematics, University of Groningen, Blauwborgje 3, 9747 AC Groningen, Netherlands
2. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona
3. 

College of Engineering, Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, EX4 4QF, Exeter

Received  September 2009 Revised  March 2010 Published  July 2010

This paper focuses on the parametric abundance and the 'Cantorial' persistence under perturbations of a recently discovered class of strange attractors for diffeomorphisms, the so-called quasi-periodic Hénon-like. Such attractors were first detected in the Poincaré map of a periodically driven model of the atmospheric flow: they were characterised by marked quasi-periodic intermittency and by $\Lambda_1>0,\Lambda_2\approx0$, where $\Lambda_1$ and $\Lambda_2$ are the two largest Lyapunov exponents. It was also conjectured that these attractors coincide with the closure of the unstable manifold of a hyperbolic invariant circle of saddle-type.
    This type of attractor is here investigated in a model map of the solid torus, constructed by a skew coupling of the Hénon family of planar maps with the Arnol$'$d family of circle maps. It is proved that Hénon-like strange attractors occur in certain parameter domains. Numerical evidence is produced, suggesting that quasi-periodic circle attractors and quasi-periodic Hénon-like attractors persist in relatively large subsets of the parameter space. We also discuss two problems in the numerical identification of so-called strange nonchaotic attractors and the persistence of all these classes of attractors under perturbation of the skew product structure.
Keywords: normally hyperbolic invariant circles, Hénon-like attractors, basins of attraction..
Mathematics Subject Classification: Primary: 37D45, 70K43; Secondary: 37M2.
Citation: Henk W. Broer, Carles Simó, Renato Vitolo. Chaos and quasi-periodicity in diffeomorphisms of the solid torus. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 871-905. doi: 10.3934/dcdsb.2010.14.871
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