In this paper, we consider synchronization of N identical nonlinear systems unidirectionally or bidirectionally coupled with time delay. First we show, using the small-gain theorem, that trajectories of coupled strictly semi-passive systems converge to a bounded region. Next, we consider the network structure under which the synchronization error dynamics has a trivial solution at zero and derive a necessary condition for synchronization with respect to the network structure. Using these facts, we then derive sufficient conditions for synchronization of the systems in terms of linear matrix inequalities via the Lyapunov-Krasovskii functional approach. The obtained results are illustrated on networks of Lorentz systems with coupling delay.
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.
We study networks of diffusively time-delay coupled oscillatory units and we show that networks with certain symmetries can exhibit a form of incomplete synchronization called partial synchronization. We present conditions for the existence and stability of partial synchronization modes in networks of oscillatory units that satisfy a semipassivity property and have convergent internal dynamics. In the study of synchronization in oscillator networks where coupling is diffusive and allows for time-delays, the focus is on deriving conditions that guarantee synchronization of all units in the network. We considered the question what happens if full synchronization cannot be achieved. Will there be no collective behavior at all or might it be possible that partial synchronization occurs, i.e., that some, but not all, units synchronize? We show that if a network contains certain symmetries, then these symmetries identify modes of partial synchronization. We present conditions for the existence and stability of partial synchronization modes in networks of diffusive time-delay coupled oscillatory units. The results are supported by numerical simulations in several networks of diffusively time-delay coupled neural Hindmarsh-Rose oscillators.
TOSHIKI 00UCHlt and ATSUSHI W AT ANA BEtThe present paper is concerned with nonlinear systems that contain delavs inside coupled with a part of state variables, which are often the cases in practical problems, but have not been treated yet. First we introduce an extension of the Lie derivative for a difference-differential equation; then Ive derive necessary and sufficient conditions for existence ofa nonlinear feedback that linearizes the input-output behaviour ofa system and decouples it from the delayed variables simultaneously, Discussions are given for two cases: firstly when the linearizing feedback contains only current values of state variables, and secondly when the linearizing feedback has memories to utilize the past values as well as the current values of state variables.
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