Two-Parameter Discontinuity-Induced Bifurcations of Limit Cycles: Classification and Open Problems

Abstract

This paper proposes a strategy for the classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems of ordinary differential equations. Such nonsmooth transitions (also known as C-bifurcations) occur when the cycle interacts with a discontinuity boundary of phase space in a nongeneric way, such as grazing contact. 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 neighborhood 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 grazing cycle is itself degenerate (e.g. nonhyperbolic) 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 with 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.

Bifurcations of the Global Stable Set of a Planar Endomorphism Near a Cusp Singularity

“…For example, piecewisesmooth systems are characterised by a critical locus where the flow or map changes in a non-smooth or discontinuous way. The study of periodic orbits of a piecewise-smooth system interacting with such a critical locus is very similar to what we have done here; see [Kuznetsov et al, 2003, Kowalczyk et al, 2006. Furthermore, the study of how the geometry is organised by such discontinuities has a similar flavour [Chillingworth, 2002].…”

Two-Parameter Bifurcation Analysis of the Buck Converter

“…It is worth mentioning here that the circuit studied in this paper serves as an excellent representative example to highlight the presence in piecewise smooth dynamical systems of a novel class of two-parameter DIB, whose occurrence was conjectured in [21,8]. According to the classification presented therein, we show that the buck converter under investigation exhibits two fundamental types of two-parameter bifurcation phenomena: those where a smooth bifurcation curve (e.g., fold or flip) merges with curves of DIBs, and those characterized by the simultaneous occurrence of more than one DIB.…”

“…As a telling example, observe the three codimension-two points, P 1 , P 2 , and P 3 , along the boundary-intersection crossing curve in the center of Figure 8. The importance of these codimension-two points was highlighted in [21], where they were classified as type II codimension-two DIBs, namely as DIBs of nonhyperbolic limit cycles.…”

Border Collision of Non-Hyperbolic Fixed Points

“…In the domain of smooth dynamical systems, the unfolding of the most common codimensiontwo points is well known (see, e.g., [Kuznetsov, 2004]), and this knowledge is exploited in continuation software for the automatic switching of bifurcation branches at these points (see, e.g., [Dhooge et al, 2002, Meijer et al, 2009). With the advent of new continuation packages for nonsmooth systems Kuznetsov, 2005, Thota andDankowicz, 2008], new effort is required to extend these results to the discontinuous case (see [di Bernardo et al, 2008, Kowalczyk and di Bernardo, 2005, Kowalczyk et al, 2006 for an overview of the subject). In this paper, we collect some results on border collision bifurcations of non-hyperbolic fixed points.…”