AUTHOR=Hittmeyer Stefanie , Krauskopf Bernd , Osinga Hinke M. , Shinohara Katsutoshi 

TITLE=Boxing-in of a blender in a Hénon-like family

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 9 - 2023

YEAR=2023

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2023.1086240

DOI=10.3389/fams.2023.1086240

ISSN=2297-4687

ABSTRACT=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.