Narrow Results

This paper investigates robust exponential stability for discrete-time recurrent neural networks with both time-varying delay (0 ≤ τ_m ≤ τ(k) ≤ τ_M) and distributed one. Through partitioning delay intervals [0,τ_m] and [τ_m,τ_M], respectively, and choosing an augmented Lyapunov-Krasovskii functional, the delay-dependent sufficient conditions are obtained by using free-weighting matrix and convex combination methods. These criteria are presented in terms of linear matrix inequalities (LMIs) and their feasibility can be easily checked by resorting to LMI in Matlab Toolbox in Ref. 1. The activation functions are not required to be differentiable or strictly monotonic, which generalizes those earlier forms. As an extension, we further consider the robust stability of discrete-time delayed Cohen-Grossberg neural networks. Finally, the effectiveness of the proposed results is further illustrated by three numerical examples in comparison with the reported ones.

We discuss synchronization in networks of Hindmarsh-Rose neurons that are interconnected via gap junctions, also known as electrical synapses. We present theoretical results for interactions without time-delay. These results are supported by experiments with a setup consisting of sixteen electronic equivalents of the Hindmarsh-Rose neuron. We show experimental results of networks where time-delay on the interaction is taken into account. We discuss in particular the influence of the network topology on the synchronization.

Recent advances in complex network research have stimulated increasing interests in understanding the relationship between the topology and dynamics of complex networks. Based on the theory of complex networks and computer simulation, we analyze the robustness to time-delay in linear consensus problem with different network topologies, such as global coupled network, star network, nearest-neighbor coupled network, small-world network, and scale-free network. It is found that global coupled network, star network, and scale-free network are vulnerable to time-delay, while nearest-neighbor coupled network and small-world network are robust to time-delay. And it is found that the maximum node degree of the network is a good predictor for time-delay robustness. And it is found that the robustness to time-delay can be improved significantly by a decoupling process to a small part of edges in scale-free network.

We introduce a time-delay function in bulk viscosity cosmology. Even for bulk viscosity functions where closed-form solutions are known, because of the time-delay term, the exact solutions are lost. Therefore in order to study the cosmological evolution of the resulting models we perform a detailed analysis of the stability of the critical points, which describe de Sitter solutions, by using Lindstedt’s method. We find that for the stability of the critical points it depends also on the time-delay parameter, where a critical time-delay value is found which plays the role of a bifurcation point. For time-delay values near the critical value, the cosmological evolution has a periodic evolution, this oscillating behavior is because of the time-delay function. We find a new behavior near the exponential expansion point, which can be seen also as an alternative way to exit the exponential inflation.

In this paper, we propose a new robust stability analysis method for uncertain cellular neural networks with time-varying delay. The proposed stability criterion is based on the Lyapunov function with sector bounded nonlinear function. The sufficient condition for the stability is derived in terms of LMI (linear matrix inequality). Numerical examples show the effectiveness of the proposed method.

In this letter, we propose a new weight learning algorithm, called an learning law (HLL), for recurrent neural networks with time-delay. Based on the Lyapunov–Krasovskii stability theory, the HLL is presented to not only guarantee asymptotical stability but also reduce the effect of external disturbance to an norm constraint. An existence condition for the HLL is represented in terms of linear matrix inequality (LMI). An illustrative example is given to demonstrate the effectiveness of the proposed HLL.

The lag-generalized synchronization of coupled time-delay chaotic systems with unknown parameters and stochastic perturbation is investigated. Based on the LaSalle-type invariance principle of stochastic differential equation, the synchronization is realized by analyzing stochastic stability of the error system. In order to achieve the synchronization, the unknown parameter update laws and the control laws are proposed. At last, two numerical examples are presented to show the effectiveness of the obtained theoretical results.

In this paper, a fractional-order (and an integer-order) chaotic system, obtained from Chua’s circuit by substituting Chua’s diode with two active coupled memristors (MRs) characterized by quadratic nonlinearity, is introduced to probe the memristive coupling effect. Two MRs connected in parallel are coupled by the flux. For the integer-order memristive system, the dynamical characteristics depending on the coupling strength coefficient between MRs without changing the circuit parameters are illustrated theoretically and numerically by using phase portraits, time domain diagram, bifurcation diagram and the Lyapunov diagram. Then based on the Adams–Bashforth–Moulton algorithm, the study of dynamic behavior of the fractional-order memristive system containing the time-delay reveals that appropriately setting the coupling strength between MRs generates more interesting attractors that differ from its integer-order counterpart. Besides, the effects of the order and the time-delay on dynamics are discussed briefly. Finally, the simulation results verify the validity of the theoretical analysis.

In this paper, a time-delayed chaos control method based on repetitive learning is proposed. A general repetitive learning control structure based on the invariant manifold of the chaotic system is given. The integration of the repetitive learning control principle and the time-delayed chaos control technique enables adaptive learning of appropriate control actions from learning cycles. In contrast to the conventional repetitive learning control, no exact knowledge (analytic representation) of the target unstable periodic orbits is needed, except for the time delay constant, which can be identified via either experiments or adaptive learning. The controller effectively stabilizes the states of the continuous-time chaos on desired unstable periodic orbits. Simulations on the Duffing and Lorenz chaotic systems are provided to verify the design and analysis.

Transition from excitability to asymptotic periodicity in an excitable system, modeled by the FitzHugh–Nagumo equations, with multiple time-delays, is analyzed. It is demonstrated that, for intermediate time-lags, the system has two coexisting attractors, a hyperbolic stable fixed point and a stable limit cycle. The fixed point is destabilized via subcritical Hopf bifurcation for much larger values of the time-lags.

It has been known that very complex chaotic behaviors can be observed in a simple first-order system with time-delay. This paper presents a fuzzy model-based approach for synchronization of time-delayed chaotic system via a scalar output variable. Takagi–Sugeno (T–S) fuzzy model can represent a general class of nonlinear system and we employ it for fuzzy modeling of the chaotic drive and response system with time-delay. Since only a scalar output variable is available for synchronization, a fuzzy observer based on T–S fuzzy model is designed and applied to chaotic synchronization. We analyze the stability of the overall fuzzy synchronization system by applying Lyapunov–Krasovskii theory and derive stability conditions by solving linear matrix inequalities (LMI's) problem. A numerical example is given to demonstrate the validity of the proposed synchronization approach.

Synchronization patterns in chains of N bi-directionally delayed coupled systems with delayed feedback are studied. Each system is hyperchaotic when decoupled from the chain. It is shown that chains with odd or even number of cites N display different spatial patterns of stable exact synchronization. When N is odd the only stable pattern of exact synchronization is between all of the units. When N is even, next to the nearest neighbors could become exactly synchronized, with the dynamics of the nearest neighbors related in a more complicated way. Sufficiently strong coupling leads to the nearest neighbor synchronization also for even N. No other patterns have been observed.

In this paper, a new approach to analyze the asymptotic, exponential and robust stability of the master-slave synchronization for Lur'e systems using time-varying delay feedback control is proposed. The discussion is motivated by the problem of transmitting information in optical communication systems using chaotic lasers. The approach is based on the Lyapunov–Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique with the use of a recent Leibniz–Newton model based transformation, without including any additional dynamics. Using the problem of synchronizing coupled Chua's circuits, three examples are given to illustrate the effectiveness of the proposed methodology.

This paper deals with the problems of stability analysis and control synthesis with performance for the master-slave synchronization of Lur'e systems using a time-delay feedback control. The proposed approach consists in obtaining a new improved robust stability criteria as well as a control law based on linear matrix inequalities (LMIs) via discretized Lyapunov–Krasovskii functional combined with an alternative strategy based on the introduction of slack variables to allow the decoupling of the system matrices from the Lyapunov matrices. Using the problem of synchronizing coupled Chua's circuits, two examples are presented to illustrate the effectiveness of the proposed methodology.

This paper deals with master–slave synchronization for Lur'e systems subject to a more general sector condition by using time delay feedback control. A new Lyapunov–Krasovskii functional and a new Lur'e–Postnikov Lyapunov functional are proposed to obtain some new delay-dependent synchronization criteria, which are formulated in the form of linear matrix inequalities (LMIs). These criteria cover some existing results as their special cases. An example shows that the result derived in this paper significantly improves some existing ones.

In the current paper, a class of stochastic cellular neural networks with reaction–diffusion effects, both discrete and distributed time delays, is studied. Several sufficient conditions guaranteeing the almost sure and pth moment exponential stability of its equilibrium solution are respectively obtained through analytic methods such as employing Lyapunov functional, applying Itô's formula, inequality techniques, embedding in Banach space, Matrix analysis and semimartingale convergence theorem. The yielded conclusions, which are independent of diffusion terms and delays, assume much less restrictions on activation functions and interconnection weights, and can be applied within a broader range of neural networks. Moreover, through the obtained results, it could be noted that noise will affect the exponential stability of the system. For illustration, two examples are given to show the feasibility of results.

This paper considers the synchronization problem for nonlinear systems with time-delay couplings. We assume that the error dynamics can be rewritten as a feedback connection of a linear delay system with multiple inputs and outputs and nonlinear elements which are decentralized and satisfy a sector condition. Then, we derive a synchronization condition for time-delay coupled systems by applying the multivariable circle criterion. Unlike the conventional synchronization criteria, the derived criterion is based on a frequency-domain stability condition and avoids the use of the Lyapunov–Krasovskii approach. As a result, the condition based on the circle criterion does not contain the conservativeness caused by the Lyapunov–Krasovskii approach. The effectiveness of the proposed criterion is shown by examples of coupled Chua systems with delay coupling.

In this paper, we study the combined effect of dynamic chemical and electrical synapses in time-delay-induced phase-transition to synchrony in coupled bursting neurons. Time-delay in coupled nonlinear oscillators or in a network of coupled nonlinear oscillators has been found to be responsible for striking dynamical behaviors such as phase-flip-transitions. These phenomena lead to synchrony or out of synchrony in different oscillators of the system. Here, we show that synaptic parameters, more precisely the neurotransmitters binding time constant influences the phase-flip-transitions of the system. We discuss how the system goes to the phase-flip-transitions when both electrical and dynamic chemical synapses are taken into account. The fourth-order Hindmarsh–Rose neuronal oscillator is considered here for the study of these transitions. A discussion on the importance of these results in brain researches is given, particularly to understand the collective dynamics of bursting neurons.

In this paper, we study the existence of rank one strange attractor in time-delayed system. First, we try to develop rank one theory for delayed differential equations. Then, we consider Chen system with time-delay, the conditions under which a supercritical Hopf bifurcation occurs are given by using the normal form method and center manifold theorem. Then, we add an external periodic force as an input and observe rank one strange attractors. Finally, several numerical simulations supporting the theoretical analysis are also given.

In this paper, we investigate the effects of time-delay and diffusion on the disease dynamics in an epidemic model analytically and numerically. We give the conditions of Hopf and Turing bifurcations in a spatial domain. From the results of mathematical analysis and numerical simulations, we find that for unequal diffusive coefficients, time-delay and diffusion may induce that Turing instability results in stationary Turing patterns, Hopf instability results in spiral wave patterns, and Hopf–Turing instability results in chaotic wave patterns. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction–diffusion epidemic model, and show that time-delay has a strong impact on the pattern formation of the reaction–diffusion epidemic model.