Available online at www.sciencedirect.com Mathematics and Computers in Simulation 95 (2013) 23–38 Original article Bandcount incrementing scenario revisited and floating regions within robust chaos Viktor Avrutin, Bernd Eckstein, Michael Schanz, Björn Schenke ∗ Institute of Parallel and Distributed Systems, University of Stuttgart, Germany Received 7 July 2011; received in revised form 10 May 2013; accepted 4 June 2013 Available online 20 June 2013 Abstract When dealing with piecewise-smooth systems, the chaotic domain often does not contain any periodic inclusions, which is called “robust chaos”. Recently, the bifurcation structures in the robust chaotic domain of 1D piecewise-linear maps were investigated. It was shown that several regions of multi-band chaotic attractors emerge at the boundary between the periodic and the chaotic domain, forming complex self-similar bifurcation structures. However, some multi-band regions were observed also far away from this boundary. In this work we consider the question how these regions emerge and how they become disconnected from the boundary. © 2013 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Piecewise smooth maps; Robust chaos; Bandcount incrementing scenario; Crises bifurcations; Border collision bifurcations 1. Introduction Piecewise-smooth dynamical systems are a focus of interest since the mid-90s for several reasons. On the one hand, they are proven to represent adequate descriptions of many practical applications for many practical systems, both in the natural and in the technical world. Typical examples are given by electronic circuits with switching behavior or specific components modeled by non-smooth functions and by mechanical systems with impacts or stick-slip behavior (see for example [20,11] for references). On the other hand, it is well-known that Poincaré return maps of smooth flows showing low-dimensional chaos (including the most popular example in this area, namely the Lorenz flow) are non-smooth and even discontinuous [12,16]. Therefore, piecewise-smooth models help to understand the basic principles of the dynamic behavior and the bifurcation phenomena in smooth systems as well. It is known that non-smooth systems demonstrate several bifurcation phenomena which do not occur in smooth systems. Especially when dealing with smooth dynamical systems, a chaotic domain in the parameter space is typically interrupted by periodic inclusions (“windows”). By contrast, piecewise-smooth systems may also show robust chaos. This term was introduced by Banerjee et al. in [10] (see also [9] for additional experimental results) and refers to the situation that an infinitesimal small variation of parameters does not influence the chaotic nature of the attractors. Typically, piecewise-smooth systems show chaotic behavior without any periodic inclusions in extended regions of ∗ Corresponding author. Tel.: +49 17 62 69 70 116. E-mail addresses: Viktor.Avrutin@ipvs.uni-stuttgart.de (V. Avrutin), Bernd.Eckstein@ipvs.uni-stuttgart.de (B. Eckstein), Michael.Schanz@ipvs.uni-stuttgart.de (M. Schanz), Bjoern.Schenke@ipvs.uni-stuttgart.de (B. Schenke). 0378-4754/$36.00 © 2013 IMACS. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.matcom.2013.06.001