Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps

Abstract

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 a physical system -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 2-D discontinuous maps.

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

“…Beyond its relation with homoclinic bifurcations for flows, such type of bifurcation scenarios have been observed in more applied contexts modeled by both one and n-dimensional piecewise-smooth maps. Examples of such applications where these bifurcation scenarios appear are power electronics ( [45,29,113,67,136,7,85,134]), control theory ( [46,58,51,133,135,50,51]), economics ( [126]) or neuroscience ( [57,74,84,89,103,122,116,124,125,123]). Typically, these bifurcation scenarios are numerically observed in two-dimensional parameter spaces near codimension-two bifurcation points.…”

“…Often, due to the complexity of real applications, one-dimensional maps are not enough and one needs to consider maps in higher dimensions. Examples of two (or higher)-dimensional piecewise-smooth maps are found as Poincaré (or stroboscopic) maps in non-autonomous mechanical systems ( [37,109,44]), but their are also found in power electronics ( [45,29,113,67,136,7]), control theory ( [46,133,135,51]) or mathematical-neuroscience or biology ( [124,125,57,103,123,116]). Piecewise-smooth maps in higher dimensions are also obtained when studying Filippov flows in R n (n ≥ 3) ( [49]); they appear as half-return Poincaré maps, and become discontinuous close to grazing and Hopf bifurcations ( [56,108,94,111,39,38]).…”

Analysis of Discontinuity Induced Bifurcations in a Dual Input DC-DC Converter

“…4, from which two critical observations can be made: (a) there is a sudden transition from a period-1 orbit to period-11 orbit and (b) after this first bifurcation takes place, there is a sequence of bifurcations where the periodicity increases or decreases by 1. As it will be shown later, these peculiar bifurcation phenomena are caused by the fact that the Poincaré map of the system is discontinuous [Jain & Banerjee, 2003;Rakshit et al, 2010].…”

“…Such maps also have been used to model induction motor drives with Direct Torque Control [Suto et al, 2009]. In all cases one-dimensional (1D) (or at most two-dimensional (2D)) discontinuous maps sufficed in describing the system dynamics with reasonable accuracy, and the theory for such maps is available [Jain & Banerjee, 2003;Rakshit et al, 2010].…”

A Simple Financial Market Model with Heterogeneous Speculators: 2-dimensional Discontinuous System

“…As the work by Rakshit et al [16] demonstrated a property of the normal form map that the unstable manifolds fold at every intersection with the x-axis, and the image of every fold point is a fold point. The stable manifolds fold at every intersection with the y-axis and the preimage of every fold point is a fold point.…”