Complex/Dynamical Systems Seminar - Bernd Kraukopf and Hinka Osinga
Bernd Krauskopf and Hinka Osinga
Department of Mathematics, ³Ô¹ÏÍø 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 (³Ô¹ÏÍø of Auckland), Andy Hammerlindl (Monash ³Ô¹ÏÍø) and Katsutoshi Shinohara (Hitotsubashi ³Ô¹ÏÍø).