Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

Dana C'Julio, Bernd Krauskopf, and Hinke M. Osinga


A dynamical system given by a diffeomorphism with a three-dimensional space may have a blender, which is a hyperbolic set Λ with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in Λ weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps H on R3, which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script in the supplemental material. In contrast to earlier work, our method allows us to determine up to 211 intersection points of the respective one-dimensional stable manifold with a chosen planar section, and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of H over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of H.

PDF copy of the paper (28MB)
Matlab demo and code (1.8MB)

Created by Hinke Osinga
Last modified: Sat Sep 16 22:04:45 NZST 2023