On the robustness of chaos in dynamical systems: Theories and applications

Abstract

This paper offers an overview of some important issues concerning the robustness of chaos in dynamical systems and their applications to the real world. PACS numbers 05.45.-a, 05.45.GgChaotic dynamical systems display two kinds of chaotic attractors: One type has fragile chaos (the attractors disappear with perturbations of a parameter or coexist with other attractors), and the other type has robust chaos, defined by the absence of periodic windows and coexisting attractors in some neighborhood of the parameter space. The existence of these windows in some chaotic regions means that small changes of the parameters would destroy the chaos, implying the fragility of this type of chaos.Contrary to this situation, there are many practical applications, such as in communication and spreading the spectrum of switch-mode power supplies to avoid electromagnetic interference [15, 16], where it is necessary to obtain reliable operation in the chaotic mode, and

Robust extremes in chaotic deterministic systems

“…An average estimate of ξ = −0.251 with standard deviation 0.002 is obtained over ρ ∈ [27,66] (Fig. 3), in good agreement with (9). Note that constancy of ξ holds for the observable φ 2 also in the parameter region where the tail index for the observable φ 1 is non-robust due to lack of hyperbolicity.…”

“…This assumption is compatible with the Chaotic Hypothesis [38,39], stating that a many particle system in a stationary state can be regarded, for the purpose of computing macroscopic properties, as a smooth dynamical system with a transitive axiom-A global attractor (a version exists for fluid dynamical systems [40]). In line with this, we expect that various types of system may display robust extremes at the experimental, observable level: robust chaotic systems [5,9], systems with a robust attractor [12], many-particle and fluid dynamical systems [38,40], high-dimensional systems [41], geophysical flows [23,35]. This may also explain the robustness which is observed in Fig.…”

“…Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighbourhood of the parameter space for a given system [5], which means that chaotic behaviour is not destroyed by small parameter variations. This is a qualitative feature and is found in several systems, from neural networks [6] to (piecewise) smooth maps [7,8], see Elhadj and Sprott [9] and references therein. A related notion is that of a robust strange attractor (entailing C 1 -openness of the defining conditions).…”

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

“…Due to their structural stability, one expects systems with hyperbolic attractors to be of particular interest as generators of robust chaos for electronic communication systems, random number generators, and various forms of encryption schemes [30][31][32].…”

Chaotic behavior analysis based on corner bifurcations

“…On the other hand, an important feature to consider is that there exist two types of chaotic attractors: the fragile chaos corresponding to the disappearance of the attractors or the coexistence of another attractors when some parameter is perturbed and the robust chaos corresponding to the absence of periodic windows that could destroy the chaotic behavior of the system for small perturbations of the bifurcation parameter. In many practical applications, it is of the most importance to maintain a robust chaotic comportment as in communications, electrical engineering [21,22], so, an overview of some important issues concerning the robustness of chaos in smooth dynamic systems is given in [1,2,12]. The result obtained in [1] contradicts the conjecture that robust chaos cannot occur for smooth systems, here, the analysis concerns piecewise smooth systems submitted to corner bifurcations, our approach guaranties a robust chaotic behavior (in the meaning that for some small bifurcation parameter variations, the system behavior stays chaotic), the robustness analysis is different from those given in the previous results and is obtained directly from the proposed approach.…”