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

• Published in: Numerical algorithms Vol. 96; no. 3; pp. 1079 - 1108
• Main Authors: C’Julio, Dana, Krauskopf, Bernd, Osinga, Hinke M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 2024; Springer Nature B.V
• Subjects: Algebra; Algorithms; Computation; Computer Science; Coordinate transformations; Dimensional stability; Eigenvalues; Fixed points (mathematics); Intersections; Invariants; Isomorphism; Manifolds (mathematics); Neighborhoods; Numeric Computing; Numerical Analysis; Original Paper; Parameterization; Parameters; Theory of Computation; 37M21; Diffeomorphism; Hénon-like family; Blender; Heterodimensional cycle; 37D25; 37C05; Partial hyperbolicity; 37C29
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-024-01812-0

A dynamical system given by a diffeomorphism with a three-dimensional phase 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 R 3 , 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 as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of 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 .