On detection of multi-band chaotic attractors

Abstract

In this work, we present two numerical methods for the detection of the number of bands of a multi-band chaotic attractor. The first method is more efficient but can be applied only for dynamical systems with a continuous system function, whereas the second one is applicable for dynamical systems with a discontinuous system function as well. Using the developed methods, we investigate a one-dimensional piecewise-linear map and report for both cases of a continuous and a discontinuous system functions some new bifurcation scenarios involving multi-band chaotic attractors.

The bandcount increment scenario. I. Basic structures

“…As shown for instance in the study by Avrutin et al (2007b) for the parameter l, only three characteristic cases have to be considered, namely l 2 fK1; 0; 1g. The case of the continuous map (lZ0) is already well investigated for both periodic and chaotic domains (Maistrenko et al 1993;Nusse & Yorke 1995;Di Bernardo et al 1999;Zhusubaliyev & Mosekilde 2003;Avrutin et al 2007a). From the two cases of positive discontinuity (lZ1) and negative discontinuity (lZK1), the first one is more complex due to a greater number of codimension-3 bifurcations.…”

“…The structure of the region P ch shown in figure 4 is calculated numerically for aZK1 using a bandcounting method reported in Avrutin et al (2007a) and implemented in the software package ANT 4.669 (www.ant4669.de). As one can see, this structure is dominated by overlapping triangle-like shapes with complex interior structures.…”

“…However, in contrast to individual crisis bifurcations, complete bifurcation scenarios formed by several types of crises bifurcations in the parameter region of chaotic dynamics without periodic inclusions (denoted as robust chaos according to Banerjee et al 1998) are still insufficiently investigated. In the previous works (Avrutin et al 2007a;Avrutin & Schanz in press), the so-called bandcount (the number of bands or strongly connected components of chaotic attractors)-adding scenario is reported, which forms a complex self-similar structure in the two-dimensional parameter space. This scenario is formed by multi-band chaotic attractors (MBCAs) and occurs in the region of chaotic dynamics close to its boundary with the region of periodic dynamics.…”

The bandcount increment scenario. II. Interior structures

“…In this work, we avoid discussion about the exact mathematical definition of a band and consider it as a strongly connected (dense) component. For the techniques used for numerical detection of the number of bands (bandcount), we refer to Avrutin et al (2007a). Since the dynamical system we investigate here has a discontinuous system function, we have to use a box-counting-based algorithm for the detection of the bandcounts.…”

Local and Global Bifurcations in Three-Dimensional, Continuous, Piecewise Smooth Maps

“…The tongue is bounded by the period-doubling bifurcation curve {θ − R 2 L 2 RL 2 : λ = −1} from right. From above, the tongue is bounded by the border collision bifurcation curve ξ 3, R 2 L 2 RL 2 and from below by the border collision bifurcation curve ξ 1,r R 2 L 2 RL 2 [Avrutin et al, 2007;Avrutin et al, 2008aAvrutin et al, , 2008b. Hence, the period-7 tongue is bounded by the border collision bifurcation curves, period-doubling bifurcation curve and Poincaré equator collision bifurcation curve.…”