A synopsis of the non-invertible, 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 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.Comment: 11 figure

Multi-parametric bifurcations in a piecewise-linear discontinuous map

In this paper a one-dimensional piecewise linear map with discontinuous system function is investigated. This map actually represents the normal form of the discrete-time representation of many practical systems in the neighbourhood of the point of discontinuity. In the 3D parameter space of this system we detect an infinite number of co-dimension one bifurcation planes, which meet along an infinite number of co-dimension two bifurcation curves. Furthermore, these curves meet at a few co-dimension three bifurcation points. Therefore, the investigation of the complete structure of the 3D parameter space can be reduced to the investigation of these co-dimension three bifurcations, which turn out to be of a generic type. Tracking the influence of these bifurcations, we explain a broad spectrum of bifurcation scenarios (like period increment and period adding) which are observed under variation of one control parameter. Additionally, the bifurcation structures which are induced by so-called big bang bifurcations and can be observed by variation of two control parameters can be explained

Multiple attractors in grazing-sliding bifurcations in Filippov type flows

We describe two examples of three-dimensional Filippov-type flows in which multiple attractors are created by grazing–sliding bifurcations. To the best of our knowledge these are the first examples to show multistability due to a grazing–sliding bifurcation in flows. In both examples, we identify the coefficients of the normal form map describing the bifurcation, and use this to find parameters with the desired behaviour. In the first example this can be done analytically, whilst the second is a dry-friction model and the identification is numerical. This explicit correspondence between the flows and a truncated normal form map reveals an important feature of the sensitivity of the predicted dynamics: the scale of the variation of the bifurcation parameter has to be very carefully chosen. Although no detailed analysis is given, we believe that this may indicate a much greater sensitivity to parameters than experience with smooth flows might suggest. We conjecture that the grazing–sliding bifurcations leading to multistability remained unreported in the literature due to this sensitivity to parameter variations