Renormalisation of the two-dimensional border-collision normal form

TL;DR

This paper investigates the complex bifurcation structures of the two-dimensional border-collision normal form in the robust chaos region using renormalisation, revealing new insights into the attractor's connected components.

Contribution

It introduces a renormalisation approach to analyze the bifurcation structure of the border-collision normal form, highlighting previously undescribed features.

Findings

Partitioned the chaos region by attractor connected components

Revealed new bifurcation structures in the map

Provided a succinct description of complex dynamics

Abstract

We study the two-dimensional border-collision normal form (a four-parameter family of continuous, piecewise-linear maps on $\mathbb{R}^2$) in the robust chaos parameter region of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049--3052, 1998]. We use renormalisation to partition this region by the number of connected components of a chaotic Milnor attractor. This reveals previously undescribed bifurcation structure in a succinct way.