A Synopsis of the Noninvertible, Two-Dimensional, Border-Collision Normal Form with Applications to Power Converters

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 noninvertible case where the two pieces fold onto the same half-plane. Most significantly we identify parameter regimes for the occurrence of key bifurcation structures, such as period-incrementing, period-adding, and robust chaos. We characterize the simplest and most dominant bifurcations and illustrate various dynamical possibilities such as invariant circles, two-dimensional attractors, and several cases of coexisting attractors. 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 noninvertible 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 model can only mimic period-doubling, although it can be subcritical.

Automated summary

The paper presents the border-collision normal form, which is a canonical form for two-dimensional, continuous maps consisting of two affine pieces. It focuses on the dynamics of this family of maps in the noninvertible case where the two pieces fold onto the same half-plane. The paper identifies parameter regimes for key bifurcation structures and explores various dynamical possibilities, and applies the results to a classic model of a boost converter.