For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Two of such consequences are the following. We also derive some consequences from this result for C1-generic dynamics (in any di-mension). Therefore, in dimension three, every heterodimensional cycle generates robust cycles. We prove that any co-index one heterodimensional cycle associated to a pair of hyperbolic saddles generates C1-robust heterodimensioal cycles. This cycle is robust if, for every g close to f, the continuations of Λ and Σ for g have a heterodimensional cycle. This cycle has co-index one if index (Λ) = index (Σ) ± 1. We study this low-frequency variability of the wind-driven, double-gyre circulation in mid-latitude ocean basins, via the bifurcation sequence that leads from steady statesĪ diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets Λ and Σ having different indices (dimension of the unstable bundle) such that the unstable manifold of Λ meets the stable one of Σ and vice-versa. Changes in this double-gyre circulation occur from year to year and decade to decade. The boundary currents and eastward jets carry substantial amounts of heat and momentum, and thus contribute in a crucial way to Earth’s climate, and to changes therein. The two gyres share the eastward extension of western boundary currents, such as the Gulf Stream or Kuroshio, and are induced by the shear in the winds that cross the respective ocean basins.
In doing so, we concentrate on the large-scale, winddriven flow of the mid-latitude oceans, which is dominated by the presence of a larger, anticyclonic and a smaller, cyclonic gyre. To illustrate the first point, we review recent theoretical advances in studying the winddriven circulation of the oceans. The purpose of this review-and-research paper is twofold: (i) to review the role played in climate dynamics by fluid-dynamical models and (ii) to contribute to the understanding and reduction of the uncertainties in future climate-change projections.