The two-dimensional border-collision normal form with a zero determinant

TL;DR

This paper explores the complex dynamics of a two-dimensional border-collision normal form with a zero determinant, revealing various bifurcation phenomena and providing insights for analyzing border-collision bifurcations in applied models.

Contribution

It characterizes the dynamics and bifurcation structures of the zero-determinant case, which is less understood, and illustrates applications to epidemic and friction oscillator models.

Findings

Identification of parameter regions with different bifurcation behaviors

Characterization of mode-locking and chaotic attractors

Discovery of three novel bifurcation structures

Abstract

The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterise the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analysed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also describe three novel bifurcation structures that remain to be explored.