Interacting global invariant sets in a planar map model of wild chaos (Q2871364)

The paper is concerned with the study of a robust singular attractor of an \(n\)-dimensional vector field, with \(n\geq5\), considered by \textit{R. Bamón}, \textit{J. Kiwi} and \textit{J. Rivera-Letelier} [``Wild Lorenz like attractors'', preprint \url{arXiv math/0508045}]. These attractors are called Lorenz-like because they are ``geometric Lorenz attractors'' in higher dimensions. In the cited paper, the \(n\)-dimensional vector field is reduced to the dynamics of an \(\left( n-1\right) \)-dimensional invertible Poincaré return map to a suitable section that has an \(\left( n-3\right) \)-dimensional strong stable foliation. The resulting quotient map is a two-dimensional noninvertible map which has a critical point that corresponds to the stable manifold of an equilibrium of the vector field. Interactions with the critical point of the two-dimensional map correspond to homoclinic or heteroclinic bifurcations in the \(n\)-dimensional vector field.NEWLINENEWLINEThe authors discuss in detail the Lorenz-like construction of the vector field, the reduction to the two-dimensional map, and the correspondences between bifurcations of the map and those of the vector field. Even though the construction of this attractor for \(n\geq5\) is similar to that of the geometric Lorenz attractor in three dimensions, it exhibits a more complicated dynamics known as wild chaos. The bifurcations of the forward critical, backward critical, stable, and unstable sets that are involved in the transition between the nonchaotic parameter regime and that of the existence of a wild Lorenz attractor are studied.NEWLINENEWLINEA thorough examination of the four types of bifurcations, the homoclinic tangency, the forward critical tangency, the backward critical tangency, and the forward-backward critical tangency indicates that they all appear consecutively in infinite sequences that accumulate on each other. In the beginning, there is an initial sequence of bifurcations that starts with a first homoclinic tangency and consists of an infinite sequence of homoclinic and an infinite sequence of forward critical tangencies accumulating on each homoclinic tangency. These are followed by a subsequent sequence of bifurcations that starts with a backward critical tangency and consists of an infinite sequence of backward critical and an infinite sequence of forward-backward critical tangencies accumulating on each backward critical tangency. The authors assert that these infinitely many homoclinic tangencies of the hyperbolic set accumulate on each other in such a dense way that they are robust.