Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

• Published in: Journal of difference equations and applications Vol. 29; no. 9-12; pp. 1094 - 1126
• Main Author: Simpson, D. J. W.
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02-12-2023; Taylor & Francis Ltd
• Subjects: Algorithms; border-collision; Invariants; Liapunov exponents; Lyapunov exponent; one-sided directional derivative; Orbits; symbol sequence; Topology; trapping region
• ISSN: 1023-6198; 1563-5120
• DOI: 10.1080/10236198.2022.2070009

We show how to formally identify chaotic attractors in continuous, piecewise-linear maps on
. For such a map f, this is achieved by constructing three objects. First,
is trapping region for f. Second,
is a finite set of words that encodes the forward orbits of all points in
. Finally,
is an invariant expanding cone for derivatives of compositions of f formed by the words in
. The existence of
,
, and C implies f has a topological attractor with a positive Lyapunov exponent. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of
and C. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large region of parameter space. We also observe how the failure of C to be expanding can coincide with a bifurcation of f. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where f is not differentiable) can be included in the analysis.