Published: Oct. 3, 2019

Bernd Krauskopf and Hinka Osinga

Department of Mathematics, University of Auckland

Hetero-dimensional Cycles and Blenders

Recent theoretical work on partially hyperbolic systems by Bonatti and Diaz (and others) has shown that chaotic dynamics may occur C1-robustly in diffeomorphisms of dimension at least three. More specifically, the existence of hetero-dimensional cycles — pairs of heteroclinic connections between two saddle periodic orbits of different index — is a C1-robust property. This result has been proved via the related concept of a blender, which is a hyperbolic set with the characterising feature that its invariant manifolds behave as geometric objects of a dimension that is larger than expected.

We consider here the question of how one can identify, characterise, and also visualise hetero-dimensional cycles and blenders. In particular, this requires concrete example systems as well as state-of-the-art numerical methods for the computation of global invariant manifolds. Firstly, we present a hetero-dimensional cycle in a four-dimensional vector field modeling intracellular calcium dynamics. We compute global invariant manifolds of two periodic orbits and show how they intersect in a connecting orbit of codimension one and an entire cylinder of connecting orbits. We present different projections of the four-dimensional phase space, as well as intersection sets in a three-dimensional Poincaré section. Secondly, we introduce an explicit Hénon-like family of three-dimensional diffeomorphisms and show that its hyperbolic set is a blender over a surprisingly large parameter range. To this end, we compute stable and unstable manifolds of fixed points in a compactified phase space to very large arclength, which allows us to check and illustrate a required denseness called the carpet property. Moreover, we discuss how the carpet property disappears.

This is joint work with Stefanie Hittmeyer and Gemma Mason (University of Auckland), Andy Hammerlindl (Monash University) and Katsutoshi Shinohara (Hitotsubashi University).