Anticipating synchronization of chaotic Lur’e systems

Abstract

In this paper we consider the anticipating synchronization of chaotic time-delayed Lur'e-type systems in a master-slave setting. We introduce three scenarios for anticipating synchronization, and give sufficient conditions for the existence of anticipating synchronizing slave systems in terms of linear matrix inequalities. The results obtained are illustrated on a time-delayed Rössler system and a time-delayed Chua oscillator. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2710964͔In their ground breaking 1990 paper, Pecora and Carroll 1 showed that, in spite of sensitive dependence on initial conditions, coupled chaotic systems can synchronize. Following this work, in the last decades considerable interest in the notion of synchronization of complex or chaotic systems has arisen. Among others, perhaps the main motivation so far seems to lie in potential applications regarding secure or private communications, though the number of reported applications is still limited to date. On the other hand synchronization appears in numerous different ways, most notably in biological applications (heart beat, brain activity, neural activity, walking, etc.), nature (planetary motion), physics (organ pipes, etc.) and engineering (coordinated robot motion, etc.). 2-4 More recently, it was observed by Voss 5 that coupled chaotic Ikeda time-delay systems can exhibit what is called anticipating synchronization in that the state of one of the systems synchronizes with the future state of the other system, which allows one to predict the state of the latter system in spite of the inherent unpredictability of chaotic systems. In many applications where synchronization seems to be essential, the coupling between systems naturally contains a small amount of delay that nevertheless does not prevent synchronization from occurring. In this regard one may think about transport delay (e.g., chemical transport between cells in a living organism) or other forms of delay like optical delay in the synchronized movement of swarms of birds. Another example occurs when multiple agents, like, e.g., robots, share a communication network, which in principle will lead to a natural delay in the execution of (shared) tasks. It is therefore of interest to investigate how anticipating synchronization can be achieved for general classes of complex or chaotic systems. In this paper we will consider a wide class of chaotic time-delay systems, so-called Lur'e systems, and discuss when and how one can design anticipating synchronization schemes for these systems.

Adaptive second-order leader-following consensus of nonlinear multi-agent systems with time-varying delay

“…By the model of the time-delayed Chua's oscillator in [21,9], the nonlinear function f (t, ·, ·) is given as therefore ρ 1 = 1.953,ρ 2 = 33.53, by Theorem 1 and Remark 2, we can obtain τ max = 0.0813, here we take τ (t) = 0.08sin(2.5t), it is easy to derive τ = 0.08 and μ = 0.2. For given k 0 = 20 and α = −0.01, by using LMI Toolbox, we can obtain δ 1 = 9.8401, δ 2 = 9.8220,δ 3 = 9.6733,δ 4 = 10.0399, δ 5 = 9.4668,δ 6 = 9.8661,δ 7 = 9.8279, δ 8 = 9.8679,δ 9 = 9.6738,δ 10 = 9.8097, the evolution of positions and velocities of ten agents are shown in Figures 2-3.…”

Pinning synchronization of delayed dynamical networks via periodically intermittent control

“…The dynamics of the Chua oscillator is given by 27,28 Figure 1 shows the chaotic behavior of the Chua attractor.…”

Characterizing the anticipating chaotic synchronization of RCL-shunted Josephson junctions

“…Such synchronization regime describes the remarkable phenomenon that it is possible that the slave dynamics act as a predictor of the master dynamics in spite of the inherent unpredictability of chaotic systems [23]. Recently, the phenomenon of anticipating synchronization has been theoretically demonstrated and experimentally vindicated in disparate dynamical systems [24][25][26][27][28]. Although there is extensive work on synchronization of coupled Josephson junctions, studies on chaotic anticipating synchronization is much less.…”