Interacting Global Invariant Sets in a Planar Map Model of Wild Chaos

Abstract

Wild chaos is a type of chaotic dynamics that can arise in a continuous-time dynamical system of dimension at least four. We are interested in the possible bifurcations or sequence of bifurcations that generate this type of chaos in dynamical systems. We focus our investigation on a planar noninvertible map introduced by Bamón, Kiwi, and Rivera-Letelier [Wild Lorenz-Like Attractors, arXiv 0508045, 2006] to prove the existence of wild chaos in a five-dimensional Lorenz-like system. The map opens up the origin (the critical point) to a disk and wraps the plane twice around it; points inside the disk have no preimage. The bounding critical circle and its images, together with the critical point and its preimages, form the so-called critical set. This set interacts with the stable and unstable sets of a saddle fixed point. Advanced numerical techniques enable us to study how the stable and unstable sets change as a parameter is varied along a path towards the wild chaotic regime. We find four types of bifurcations: The stable and unstable sets interact with each other in homoclinic tangencies (which also occur in invertible maps), and they interact with the critical set in three types of bifurcations specific to noninvertible maps, which cause changes (such as self-intersections) of the topology of these global invariant sets. Overall, a consistent sequence of all four types of bifurcations emerges, which we present as a first attempt towards explaining the geometric nature of wild chaos. Using two-parameter bifurcation diagrams, we show that essentially the same sequences of bifurcations occur along different paths towards the wild chaotic regime, and we use this information to obtain an indication of the size of the parameter region where wild chaos exists.