Two-parameter nonsmooth grazing bifurcations of limit cycles: classification and open problems

This paper proposes a strategy for the classification of codimension-two grazing bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a non-generic way. Several such codimension-one events have recently been identified, causing for example period-adding or sudden onset of chaos. Here, the focus is on codimension-two grazings that are local in the sense that the dynamics can be fully described by an appropriate Poincaré map from a neighbourhood of the grazing point (or points) of the critical cycle to itself. It is proposed that codimension-two grazing bifurcations can be divided into three distinct types: either the grazing point is degenerate, or the the grazing cycle is itself degenerate (e.g. non-hyperbolic) or we have the simultaneous occurrence of two grazing events. A careful distinction is drawn between their occurrence in systems with discontinuous states, discontinuous vector fields, or that have discontinuity in some derivative of the vector field. Examples of each kind of bifurcation are presented, mostly derived from mechanical applications. For each example, where possible, principal bifurcation curves characteristic to the codimension-two scenario are presented and general features of the dynamics discussed. Many avenues for future research are opened.

Nonsmooth bifurcations in a piecewise-linear model of the Colpitts oscillator

This paper deals with the implications of considering a first-order approximation of the circuit nonlinearities in circuit simulation and design. The Colpitts oscillator is taken as a case study and the occurrence of discontinuous bifurcations, namely, border-collision bifurcations, in a piecewise-linear model of the oscillator is discussed. In particular, we explain the mechanism responsible for the dramatic changes of dynamical behavior exhibited by this model when one or more of the circuit parameters are varied. Moreover, it is shown how an approximate one-dimensional (1-D) map for the Colpitts oscillator can be exploited for predicting border-collision bifurcations. It turns out that at a border collision bifurcation, a 1-D return map of the Colpitts oscillator exhibits a square-root-like singularity. Finally, through the 1-D map, a two-parameter bifurcation analysis is carried out and the relationships are pointed out between border-collision bifurcations and the conventional bifurcations occurring in smooth system

Complex behavior in switching power converters

Author name used in this publication: Chi K. Tse2001-2002 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Simultaneous Border-Collision and Period-Doubling Bifurcations

We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely classified; in particular, we give conditions that ensure chaos.Comment: 22 pages; 5 figure

Bifurcation Phenomena in Two-Dimensional Piecewise Smooth Discontinuous Maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border, and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, many switching dynamical systems have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. The theory for understanding the bifurcation phenomena in such systems is not available yet. In this paper we present the first approach to the problem of analysing and classifying the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We explain the bifurcations occurring in the static VAR compensator used in electrical power systems, using the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps

Dynamics of a piecewise smooth map with singularity

Experiments observing the liquid surface in a vertically oscillating container have indicated that modeling the dynamics of such systems require maps that admit states at infinity. In this paper we investigate the bifurcations in such a map. We show that though such maps in general fall in the category of piecewise smooth maps, the mechanisms of bifurcations are quite different from those in other piecewise smooth maps. We obtain the conditions of occurrence of infinite states, and show that periodic orbits containing such states are superstable. We observe period-adding cascade in this system, and obtain the scaling law of the successive periodic windows.Comment: 10 pages, 6 figures, composed in Latex2