Existence of blenders in a Hénon-like family: geometric insights from invariant manifold computations. (21st August 2018)

• Record Type: Journal Article
• Authors: Hittmeyer, Stefanie; Krauskopf, Bernd; Osinga, Hinke M; Shinohara, Katsutoshi
• Is Part Of: Nonlinearity. Volume 31:Number 10(2018:Oct.)
• Page Start: R239
• Page End: R267
• Publication Date: 2018-08-21
• Subjects: diffeomorphism -- non-uniform hyperbolicity -- global invariant manifolds -- wild chaos; 37C05 -- 37D25 -- 37D10 -- 37M20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aacd66
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: 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 Hénon-like 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.