Local analysis of \(C\)-bifurcations in \(n\)-dimensional piecewise-smooth dynamical systems
type: Item
instance of: scholarly article
title: Local analysis of \(C\)-bifurcations in \(n\)-dimensional piecewise-smooth dynamical systems
zbMATH Open document ID: 0967.37030
DOI: 10.1016/S0960-0779(98)00317-8
published in: Chaos, Solitons and Fractals
publication date: 1999-11-29T00:00:00Z
review text: Many models in engineering and applications involve piecewise-smooth, continuous or discontinuous, maps or differential equations, e.g. impact oscillators. The authors reformulate and extend a theory of local codimension one bifurcations in such \(n\)-dimensional systems, which was originated by one of the authors, \textit{M. I. Feigin}, in the 70s in Russia [Prikl. Mat. Mekh. 34, 861-869 (1970; Zbl 0224.34021), ibid. 42, 820-829 (1978; Zbl 0419.58014)], in a work on \(C\)-bifurcations (border-collision bifurcation). In the ODE formulation, for a suitable section of a periodic orbit, the authors derive a local Poincaré map, whose linearization yields conditions for typical \(C\)-bifurcations (related to period doubling). These conditions can be given in terms of the spectrum of the linearized local map which completely characterize this bifurcation scenario. For a class of two dimensional piecewise continuous maps, a classification of all codimension one bifurcations is given. More complexity is pictured for a three parameter 1-D piecewise smooth map. Finally the methods are applied to a three parameter ODE. Numerical data and figures illustrate these examples.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 70K50
zbMATH DE Number: 1371680
zbMATH Keywords: C-bifurcations
zbMATH Keywords: local map
zbMATH Keywords: piecewise-smooth system
zbMATH Keywords: non-standard bifurcation
zbMATH Keywords: border-collision bifurcation
author: Mario Di Bernardo
author: M. I. Feĭgin
author: S. J. Hogan
author: Martin E. Homer
reviewed by: Jens D. M. Rademacher
describes a project that uses: Dynamics
MaRDI profile type: Publication
full work available at URL: s0960-0779(98)00317-8
OpenAlex ID: W4246664411
description: scientific article; zbMATH DE number 1371680