Robust Chaos

Abstract

It has been proposed to make practical use of chaos in communication, in enhancing mixing in chemical processes and in spreading the spectrum of switch-mode power suppies to avoid electromagnetic interference. It is however known that for most smooth chaotic systems, there is a dense set of periodic windows for any range of parameter values. Therefore in practical systems working in chaotic mode, slight inadvertent fluctuation of a parameter may take the system out of chaos. We say a chaotic attractor is robust if, for its parameter values there exists a neighborhood in the parameter space with no periodic attractor and the chaotic attractor is unique in that neighborhood. In this paper we show that robust chaos can occur in piecewise smooth systems and obtain the conditions of its occurrence. We illustrate this phenomenon with a practical example from electrical engineering.Comment: 4 pages, Latex, 4 postscript figures, To appear in Phys. Rev. Let

The bandcount increment scenario. I. Basic structures

“…The observed dynamic behaviour represents robust chaos in the sense of Banerjee et al (1998), which means it is not interrupted by any periodic inclusions (windows). Nevertheless, system (2.1) undergoes several crises in this region, which change the topological structure of the attractors and form a complex bifurcation scenario.…”

“…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.…”

Less Is More II: An Optimistic View of Piecewise Smooth Bifurcation Theory

“…The criteria for the theorem to hold are easy to verify numerically making it possible to determine regions on which Young's Theorem holds and compare these with theoretical bounds in [1], see [3] for details. The point about this result is that one could be tempted to provide further details such as the Hausdorff dimension of the support of the measure (the attractor), but that the statement that there is an attractor with an invariant measure having a nice one-dimensional projection gives the essential picture without overcomplicating the story: less is more.…”

“…The parameter µ is considered to be the bifurcation parameter and some results for these maps are described in [4]. Banerjee et al [1] show that the border collision normal form has parameters with (a) a trapping region; and (b) transverse intersections of stable and unstable manifolds and hence quasi-one-dimensional attractors: this has been called robust chaos. Young provided the tools to make these statements more precise [6].…”

Lessons Learnt from the Cryptanalysis of Chaos-Based Ciphers

“…As a conclusion, it is highly advisable to use dynamical systems with chaotic behavior for all the values of the control parameter(s). That is, robust chaotic systems [30] should be used instead of non-robust ones.…”