Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps
type: Item
instance of: scholarly article
title: Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps
zbMATH Open document ID: 1402.37061
author: Laura Gardini
author: Roya Makrooni
author: Iryna Sushko
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2018-09-06T00:00:00Z
review text: The authors consider the map \(x_{n+1}=f(x_n)\) where \[ f(x)=\begin{cases} ax-1, & x\leq 0, \\ bx^{-\gamma}-1, & x>0. \end{cases} \] Here \(\gamma>0\) so that the map is discontinuous. If \(a<0,b<0\) the map is invertible and if \(a<0, b>0\) it is noninvertible. A thorough, although not complete, description of the dynamics is given. Bifurcations of all fixed points are described. In the invertible case certain period-2 and period-4 orbits are shown to be involved in both smooth (i.e., occurring in smooth dynamical systems) and border collision bifurcations. These orbits are attracting for some parameter values. In the noninvertible case, for \(\gamma>1\) there is shown to be a sequence of alternating smooth period-doubling bifurcations and border collision bifurcations leading to attracting orbits of period \(n,2n,4n-1,2(4n-1),4(4n-1)-1,\dots\). For \(0<\gamma<1\) there may be unbounded chaotic attractors, or an attracting periodic orbit, or coexistence of a stable fixed point and an unbounded chaotic attractor.
Mathematics Subject Classification ID: 37G35
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 37N30
Mathematics Subject Classification ID: 39A28
zbMATH DE Number: 6932759
zbMATH Keywords: piecewise smooth maps
zbMATH Keywords: border collision bifurcations
zbMATH Keywords: unbounded chaotic sets
reviewed by: Carlo Laing
MaRDI profile type: Publication
full work available at URL: dcdsb.2018039
OpenAlex ID: W2779652307
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DOI: 10.3934/DCDSB.2018039
description: scientific article; zbMATH DE number 6932759