A synopsis of the non-invertible, two-dimensional, border-collision normal form with applications to power converters

TL;DR

This paper explores the dynamics of non-invertible, two-dimensional border-collision maps, identifying key bifurcation regimes and applying findings to a boost converter model to understand its bifurcation behavior.

Contribution

It provides a comprehensive guide to non-invertible border-collision normal forms and demonstrates their application to power converter dynamics, highlighting the specific bifurcation behaviors observed.

Findings

Identified parameter regimes for bifurcations like period-incrementing and chaos.

Applied theoretical results to a boost converter model.

Found that the converter's bifurcation mimics only period-doubling, often subcritical.

Abstract

The border-collision normal form is a canonical form for two-dimensional, continuous maps comprised of two affine pieces. In this paper we provide a guide to the dynamics of this family of maps in the non-invertible case where the two pieces fold onto the same half-plane. We identify parameter regimes for the occurrence of key bifurcation structures, such as period-incrementing, period-adding, and robust chaos. We then apply the results to a classic model of a boost converter for adjusting the voltage of direct current. It is known that for one combination of circuit parameters the model exhibits a border-collision bifurcation that mimics supercritical period-doubling and is non-invertible due to the switching mechanism of the converter. We find that over a wide range of parameter values, even though the dynamics created in border-collision bifurcations is in general extremely diverse, the bifurcation in the boost converter can only mimic period-doubling, although it can be subcritical.