A blender is an intricate geometric structure of a diffeomorphism of dimension at least three. Its characterizing feature is that its invariant manifolds behave as geometric objects of a dimension that is larger than expected from the dimensions of the manifolds themselves. We introduce an explicit Hénonlike family of three-dimensional diffeomorphisms and show that it has a blender in a specific parameter regime. Advanced numerical techniques for the computation of one-dimensional stable and unstable manifolds enable us to present images of actual blenders and their associated manifolds. Moreover, these techniques allow us to present strong numerical evidence for the existence of the blender over a larger parameter range, as well as its disappearance and geometric properties beyond this range.

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 selfintersections) 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. Introduction.Vector fields and diffeomorphisms with chaotic attractors arise in numerous fields of applications, including neuroscience [23], chemical reactions [24], and laser systems [45]. One of the first and best known examples of a chaotic three-dimensional vector field is the Lorenz system [44]. It was introduced in 1963 by meteorologist Lorenz as a simplified model of convection dynamics in the earth's atmosphere. At the classical parameter values, its chaotic attractor-the Lorenz attractor -has the shape of the wings of a butterfly. The dynamics on this attractor has been studied in great detail; see, for example, [2,3,32,62,63]. More recently, the focus has shifted to what this means for the global behavior on the entire phase space [19,20].The approach taken for studying the Lorenz system at the classical parameter values is

A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

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This contribution provides a geometric perspective on the type of chaotic dynamics that one finds in the original Lorenz system and in a higher-dimensional Lorenz-type system. The latter provides an example of a system that features robustness of homoclinic tangencies; one also speaks of 'wild chaos' in contrast to the 'classical chaos' where homoclinic tangencies accumulate on one another, but do not occur robustly in open intervals in parameter space. Specifically, we discuss the manifestation of chaotic dynamics in the three-dimensional phase space of the Lorenz system, and illustrate the geometry behind the process that results in its description by a one-dimensional noninvertible map. For the higher-dimensional Lorenz-type system, the corresponding reduction process leads to a two-dimensional noninvertible map introduced in 2006 by Bamón, Kiwi, and Rivera-Letelier [arXiv 0508045] as a system displaying wild chaos. We present the geometric ingredientsin the form of different types of tangency bifurcations-that one encounters on the route to wild chaos.