Chaos in the border-collision normal form: A computer-assisted proof using induced maps and invariant expanding cones

TL;DR

This paper introduces a computer-assisted method using induced maps and invariant cones to rigorously prove the existence of chaotic attractors with positive Lyapunov exponents in a two-dimensional border-collision map, improving efficiency over traditional Lyapunov estimates.

Contribution

The paper develops an algorithm to verify conditions for chaos in border-collision maps using induced maps, enabling rigorous proofs with less computation.

Findings

Proved existence of robust chaos in a parameter region with strong rotational dynamics.

Demonstrated the effectiveness of the induced map approach for rigorous chaos verification.

Provided a computationally efficient method for analyzing attractors in piecewise-linear maps.

Abstract

In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The algorithm requires relatively few computations, so it is a more efficient method than, for example, estimating the Lyapunov exponent from a single orbit in terms of speed, numerical accuracy, and rigor. The algorithm is used to prove the existence of an attractor with a positive Lyapunov exponent numerically in an area of parameter space where the map has strong rotational characteristics and the consideration of an induced map is critical for the proof of robust chaos.