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

TL;DR

This paper presents an algorithm to detect chaos in two-dimensional piecewise-linear maps by identifying invariant expanding cones, trapping regions, and encoding orbits, enabling computer-assisted proofs of chaotic behavior.

Contribution

The authors develop a method to explicitly construct invariant cones and trapping regions, providing a new computational approach to verify chaos in piecewise-linear maps.

Findings

Existence of invariant cones correlates with chaotic dynamics.

The algorithm successfully identifies chaos in a large parameter regime.

Failure of the cone expansion indicates bifurcations.

Abstract

We show how the existence of three objects, $\Omega_{\rm trap}$, ${\bf W}$, and $C$, for a continuous piecewise-linear map $f$ on $\mathbb{R}^N$, implies that $f$ has a topological attractor with a positive Lyapunov exponent. First, $\Omega_{\rm trap} \subset \mathbb{R}^N$ is trapping region for $f$. Second, ${\bf W}$ is a finite set of words that encodes the forward orbits of all points in $\Omega_{\rm trap}$. Finally, $C \subset T \mathbb{R}^N$ is an invariant expanding cone for derivatives of compositions of $f$ formed by the words in ${\bf W}$. 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 $\Omega_{\rm trap}$ 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 regime 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.