Boxing-in of a blender in a Hénon-like family - Hittmeyer, Stefanie; Krauskopf, Bernd; Osinga, Hinke M.; Shinohara, Katsutoshi (Frontiers in Applied Mathematics and Statistics, 2023)
The extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. We consider here a Hénon-like family in R³, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed ponts with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked numerically for extremely long pieces of global one-dimensional manifolds to determine over which parameter range a blender exists in the Hénon-like family. In the work presented here, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelepiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold. This constitutes a three-dimensional horseshoe that, in contrast to the idealised affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter- dependent way, and this allows us to characterise properties of the hyperbolic set in an intuitive way. In particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.