Chaos and quasi-periodicity in diffeomorphisms of the solid torus

Abstract

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.