Synchronization of Time-Delay Coupled Neural Networks With Stabilizing Delayed Impulsive Control

This brief studies the distributed synchronization of time-delay coupled neural networks (NNs) with impulsive pinning control involving stabilizing delays. A novel differential inequality is proposed, where the state's past information at impulsive time is effectively extracted and used to handle the synchronization of coupled NNs. Based on this inequality, the restriction that the size of impulsive delay is always limited by the system delay is removed, and the upper bound on the impulsive delay is relaxed, which is improved the existing related results. By using the methods of average impulsive interval (AII) and impulsive delay, some relaxed criteria for distributed synchronization of time-delay coupled NNs are obtained. The proposed synchronization conditions do not impose on the upper bound of two consecutive impulsive signals, and the lower bound is more flexible. Moreover, our results reveal that the impulsive delays may contribute to the synchronization of time-delay systems. Finally, typical networks are presented to illustrate the advantage of our delayed impulsive control method.

Interval Bipartite Synchronization of Multiple Neural Networks in Signed Graphs

Interval bipartite consensus of multiagents described by signed graphs has received extensive concern recently, and the rooted cycles play a critical role in stabilization, while the structurally balanced graphs are essential to achieve bipartite consensus. However, the gauge transformation used in the linear system is no longer feasible in the nonlinear case. This article addresses interval bipartite synchronization of multiple neural networks (NNs) in a signed graph via a Lyapunov-based approach, extending the existing work to a more practical but complicated case. A general matrix M in signed graphs is introduced to construct the novel Lyapunov functions, and sufficient conditions are obtained. We find that the rooted cycles and the structurally balanced graphs are essential to stabilize and achieve bipartite synchronization. More importantly, we discover that the nonrooted cycles are crucial in reaching interval bipartite synchronization, not previously mentioned. Several examples are presented to illustrate interval bipartite synchronization of multiple NNs with signed graphs.

Desynchronizing oscillators coupled in multi-cluster networks through adaptively controlling partial networks

This article introduces an adaptive control scheme with a feedback delay, specifically designed for controlling partial networks, to achieve desynchronization in a coupled network with two or multiple clusters. The proposed scheme's effectiveness is validated through several representative examples of coupled neuronal networks with two interconnected clusters. The efficacy of this scheme is attributed to the rigorous and numerical analyses on the corresponding transcendental characteristic equation, which includes time delay and other network parameters. In addition to investigating the impact of time delay and inter-connectivity on the stability of an incoherent state, we also rigorously find that controlling only one cluster cannot realize the desynchronization in the coupled oscillators within three or more clusters. All these, we believe, can deepen the understanding of the deep brain stimulation techniques presently used in the clinical treatment of neurodegenerative diseases and suggest future avenues for enhancing these clinical techniques through adaptive feedback settings.

Impulsive strategies in nonlinear dynamical systems: A brief overview

The studies of impulsive dynamical systems have been thoroughly explored, and extensive publications have been made available. This study is mainly in the framework of continuous-time systems and aims to give an exhaustive review of several main kinds of impulsive strategies with different structures. Particularly, (i) two kinds of impulse-delay structures are discussed respectively according to the different parts where the time delay exists, and some potential effects of time delay in stability analysis are emphasized. (ii) The event-based impulsive control strategies are systematically introduced in the light of several novel event-triggered mechanisms determining the impulsive time sequences. (iii) The hybrid effects of impulses are emphatically stressed for nonlinear dynamical systems, and the constraint relationships between different impulses are revealed. (iv) The recent applications of impulses in the synchronization problem of dynamical networks are investigated. Based on the above several points, we make a detailed introduction for impulsive dynamical systems, and some significant stability results have been presented. Finally, several challenges are suggested for future works.

Fully Distributed Synchronization of Complex Networks With Adaptive Coupling Strengths

This article considers the fully distributed leaderless synchronization in a complex network by only utilizing local neighboring information to design and tune the coupling strength of each node such that the synchronization problem can be solved without involving any global information of the network. For an undirected network, a fully distributed synchronization algorithm is presented to adjust the coupling strength of each node based on a simple adaptive law. When the topology of a network is directed, two different types of adaptive algorithms are developed to achieve synchronization in a fully distributed manner, where the coupling strength of each node is designed to be either the sum or product of two non-negative scalar functions. The fully distributed leaderless synchronization of a directed network is investigated in a leader-follower framework, where the leader subnetwork is analyzed by using the techniques from constrained Rayleigh quotients and the follower subnetwork is addressed by employing the properties of nonsingular M -matrices. Simulations are given to illustrate the theoretical results.

Feedback Pinning Control of Successive Lag Synchronization on a Dynamical Network

In nature and human society, successive lag synchronization (SLS) is an important synchronization phenomenon. Compared with other synchronization patterns, the control theory of SLS is very lacking. To this end, we first introduce a complex dynamical network model with distributed delayed couplings, and design both the linear feedback pinning control and adaptive feedback pinning control to push SLS to the desired trajectories. Second, we obtain a series of sufficient conditions to achieve SLS to a desired trajectory with global stability. What is more, the control flow of SLS is given to show how to pick the pinned nodes accurately and set the feedback gains as well. Finally, since time-varying delay is common, we extend the constant time delay in SLS to be time varying. We find that the proposed pinning control schemes are still feasible if the coupling terms are appropriately adjusted. The theoretical results are verified on a neural network and the coupled Chua's circuits.

Quasisynchronization of Heterogeneous Neural Networks With Time-Varying Delays via Event-Triggered Impulsive Controls

Time delays are unavoidable since they are ubiquitous and may have a great impact on the performance of neural networks. Resources efficiency is a common concern in many networked systems with limited resources. This article investigates quasisynchronization of the heterogeneous neural networks with time-varying delays via event-triggered impulsive controls which combine the impulsive control and the event-triggered technique. The centralized and distributed event-triggered impulsive controls are, respectively, presented. The suitable Lyapunov functions are constructed, and the triggering functions are derived, which guarantee that not only are the synchronization errors less than a non-negative bound but also the Zeno behaviors can be eliminated. It is suggested that the distributed one has great superiority in taking up fewer resources compared with the time-triggered impulsive control. Numerical examples are proposed to verify the validity of the centralized and distributed control methods.

Quasisynchronization of Heterogeneous Dynamical Networks via Event-Triggered Impulsive Controls

The time-triggered impulsive control of complex homogeneous dynamical networks has received wide attention due to its occasional occupation of the communication channels. This article is devoted to quasisynchronization of heterogeneous dynamical networks via event-triggered impulsive controls with less channel occupation. Two kinds of triggered mechanisms, that is, the centralized event-triggered mechanism in which the control is updated based upon the state information of all nodes, and the distributed event-triggered mechanism where the control is updated according to the state information of each node and its neighboring node, are proposed, respectively, such that the synchronization error between the heterogeneous dynamical networks and a virtual target is not more than a nonzero bound. What is more, the Zeno behavior is shown to be excluded. It is found that the combination method of the event-triggered control and the impulsive control, that is, the distributed event-triggered impulsive control has the advantage of low-energy consumption and takes up many fewer communication channels over the time-triggered impulsive control. Two numerical examples are conducted to illustrate the effectiveness of the proposed event-triggered impulsive controls.

Pinning control of complex networks with time-varying inner and outer coupling

This paper addresses the pinning synchronization of nonlinear multiple time-varying coupling complex networks. Time-varying inner coupling in the single node state space and time-varying outer coupling among nodes in an entire complex network are taken into consideration. The main contribution is to propose some pinning synchronization criterion by which time-varying complex networks can be synchronized to the desired state. Besides, different parameters of linear controllers, adaptive controllers and adaptive coupling strength on the synchronization have been investigated. It is found that complex networks can achieve global synchronization by adaptively adjusting the coupling strength or controllers. Finally, simulation examples of random networks are given to verify the theoretical results.

Bipartite Synchronization of Multiple Memristor-Based Neural Networks With Antagonistic Interactions

In this article, by introducing a signed graph to describe the coopetition interactions among network nodes, the mathematical model of multiple memristor-based neural networks (MMNNs) with antagonistic interactions is established. Since the cooperative and competitive interactions coexist, the states of MMNNs cannot reach complete synchronization. Instead, they will reach the bipartite synchronization: all nodes' states will reach an identical absolute value but opposite sign. To reach bipartite synchronization, two kinds of the novel node- and edge-based adaptive strategies are proposed, respectively. First, based on the global information of the network nodes, a node-based adaptive control strategy is constructed to solve the bipartite synchronization problem of MMNNs. Secondly, a local edge-based adaptive algorithm is proposed, where the weight values of edges between two nodes will change according to the designed adaptive law. Finally, two simulation examples validate the effectiveness of the proposed adaptive controllers and bipartite synchronization criteria.

Global Exponential Synchronization of Coupled Delayed Memristive Neural Networks With Reaction-Diffusion Terms via Distributed Pinning Controls

This article presents new theoretical results on global exponential synchronization of nonlinear coupled delayed memristive neural networks with reaction-diffusion terms and Dirichlet boundary conditions. First, a state-dependent memristive neural network model is introduced in terms of coupled partial differential equations. Next, two control schemes are introduced: distributed state feedback pinning control and distributed impulsive pinning control. A salient feature of these two pinning control schemes is that only partial information on the neighbors of pinned nodes is needed. By utilizing the Lyapunov stability theorem and Divergence theorem, sufficient criteria are derived to ascertain the global exponential synchronization of coupled neural networks via the two pining control schemes. Finally, two illustrative examples are elaborated to substantiate the theoretical results and demonstrate the advantages and disadvantages of the two control schemes.

Polaritons and excitons: Hamiltonian design for enhanced coherence

The primary questions motivating this report are: Are there ways to increase coherence and delocalization of excitation among many molecules at moderate electronic coupling strength? Coherent delocalization of excitation in disordered molecular systems is studied using numerical calculations. The results are relevant to molecular excitons, polaritons, and make connections to classical phase oscillator synchronization. In particular, it is hypothesized that it is not only the magnitude of electronic coupling relative to the standard deviation of energetic disorder that decides the limits of coherence, but that the structure of the Hamiltonian-connections between sites (or molecules) made by electronic coupling-is a significant design parameter. Inspired by synchronization phenomena in analogous systems of phase oscillators, some properties of graphs that define the structure of different Hamiltonian matrices are explored. The report focuses on eigenvalues and ensemble density matrices of various structured, random matrices. Some reasons for the special delocalization properties and robustness of polaritons in the single-excitation subspace (the star graph) are discussed. The key result of this report is that, for some classes of Hamiltonian matrix structure, coherent delocalization is not easily defeated by energy disorder, even when the electronic coupling is small compared to disorder.

Scalable Distributed Filtering for a Class of Discrete-Time Complex Networks Over Time-Varying Topology

This article is concerned with the distributed filtering problem for a class of discrete complex networks over time-varying topology described by a sequence of variables. In the developed scalable filtering algorithm, only the local information and the information from the neighboring nodes are used. As such, the proposed filter can be implemented in a truly distributed manner at each node, and it is no longer necessary to have a certain center node collecting information from all the nodes. The aim of the addressed filtering problem is to design a time-varying filter for each node such that an upper bound of the filtering error covariance is ensured and the desired filter gain is then calculated by minimizing the obtained upper bound. The filter is established by solving two sets of recursive matrix equations, and thus, the algorithm is suitable for online application. Sufficient conditions are provided under which the filtering error is exponentially bounded in mean square. The monotonicity of the filtering error with respect to the coupling strength is discussed as well. Finally, an illustrative example is presented to demonstrate the feasibility and effectiveness of our distributed filtering strategy.

Fully Distributed Synchronization of Dynamic Networked Systems With Adaptive Nonlinear Couplings

In this article, we consider the distributed synchronization problem of dynamic networked systems with adaptive nonlinear couplings. Based on how the information is collected, the interactions between subsystems are characterized by nonlinear relative state couplings and nonlinear absolute state couplings. In both cases, we show that the considered nonlinear interactions can be used to simulate the couplings with disturbed relative or absolute states. In order to implement the nonlinear couplings in a fully distributed fashion, adaptive control laws are proposed for the adjustment of coupling strengths between connected subsystems. It is shown that the connected network topology is sufficient to ensure the synchronization of dynamic networked systems with the proposed adaptive nonlinear coupling methods. Different from many existing works, the σ -modification technique is used to suppress the increase of the coupling strengths with an additional benefit of preventing the coupling strengths from increasing. Simulation examples are given to assess the performance of the proposed adaptive nonlinear couplings.

Synchronization of the Networked System With Continuous and Impulsive Hybrid Communications

Many networked systems display some kind of dynamics behaving in a style with both continuous and impulsive communications. The cooperation behaviors of these networked systems with continuous connected or impulsive connected or both connected topologies of communications are important to understand. This paper is devoted to the synchronization of the networked system with continuous and impulsive hybrid communications, where each topology of communication mode is not connected in every moment. Two kind of structures, i.e., fixed structure and switching structures, are taken into consideration. A general concept of directed spanning tree (DST) is proposed to describe the connectivity of the networked system with hybrid communication modes. The suitable Lyapunov functions are constructed to analyze the synchronization stability. It is showed that for fixed topology having a jointly DST, the networked system with continuous and impulsive hybrid communication modes will achieve asymptotic synchronization if the feedback gain matrix and the average impulsive interval are properly selected. The results are then extended to the switching case where the graph has a frequently jointly DST. Some simple examples are then given to illustrate the derived synchronization criteria.

Sampled-based consensus for nonlinear multi-agent systems with average graph

In this paper, we discuss the consensus problem for a class of nonlinear multiagent systems (MASs) with sampled-data and switched topologies. Considering the topology switching among a set of disconnected graphs, we propose "average graph" based on the switching frequency. We then introduce a sampled-based consensus protocol to guarantee that the MAS network will achieve consensus. By means of the Lyapunov function and contradiction theory, we derive the consensus criterion in which the average graph of the MAS communication network contains a directed spanning tree. Moreover, the allowable upper bound of the sampling period is obtained from the Laplacian matrix of the average graph. A numerical example is, finally, given to verify the effectiveness of the theoretical results.

Avoiding Congestion in Cluster Consensus of the Second-Order Nonlinear Multiagent Systems

In order to avoid congestion in the second-order nonlinear leader-following multiagent systems over capacity-limited paths, an approach called cluster lag consensus is proposed, which means that the agents in different clusters will pass through the same positions with the same velocities but lag behind the leader at different times. Lyapunov functionals and matrix theory are applied to analyze such cluster lag consensus. It is shown that when the graphic roots of clusters are influenced by the leader and the intracoupling of cluster agents is larger than a threshold, the cluster lag consensus can be achieved. Furthermore, the cluster lag consensus with a time-varying communication topology is investigated. Finally, an illustrative example is presented to demonstrate the effectiveness of the theoretical results. In particular, when the physical sizes of the agents are taken into consideration, it is shown that with a rearrangement and a position transformation, the multiagent system will reach cluster lag consensus in the new coordinate system. This means that all agents in the same cluster will reach consensus on the velocity, but their positions may be different and yet their relative positions converge to a constant asymptotically.

Synchronization of heterogeneous linear networks with distinct inner coupling matrices

In this paper, we study synchronization of heterogeneous linear networks with distinct inner coupling matrices. Firstly, for synchronous networks, we show that any synchronous trajectory will converge to a corresponding synchronous state. Then, we provide an invariant set, which can be exactly obtained by solving linear equations and then used for characterizing synchronous states. Afterwards, we use inner coupling matrices and node dynamics to successively decompose the original network into a new network, composed of the external part and the internal part. Moreover, this new network can be proved to synchronize to the above invariant set by constructing the corresponding desired Lyapunov-like functions for the internal part and the external part respectively. In particular, this result still holds if the coupling strength is disturbed slightly. Finally, examples with numerical simulations are given to illustrate the validity and applicability of our theoretical results.

Finite-Time Synchronization of Coupled Hierarchical Hybrid Neural Networks With Time-Varying Delays

This paper is concerned with the finite-time synchronization problem of coupled hierarchical hybrid delayed neural networks. This coupled hierarchical hybrid neural networks consist of a higher level switching and a lower level Markovian jumping. The time-varying delays are dependent on not only switching signal but also jumping mode. By using a less conservative weighted integral inequality and stochastic multiple Lyapunov-Krasovskii functional, new finite-time synchronization criteria are obtained, which makes the state trajectories be kept within the prescribed bound in a time interval. Finally, an example is proposed to demonstrate the effectiveness of the obtained results.

Pinning Control of Lag-Consensus for Second-Order Nonlinear Multiagent Systems

Lag consensus means that the corresponding state vectors of the followers are behind the leader with a lag time. In this paper, Lyapunov functional and matrix theory are applied to analyze pinning-controlled lag consensus of second-order nonlinear multiagent systems. The focus is twofold: 1) to find out which agents should be pinned and 2) to determine what the coupling strength should be, so that the multiagent systems can reach lag consensus. Moreover, the practical problem in a noisy environment is considered. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed pinning control protocol.

Behavioral synchronization induced by epidemic spread in complex networks

During the spread of an epidemic, individuals in realistic networks may exhibit collective behaviors. In order to characterize this kind of phenomenon and explore the correlation between collective behaviors and epidemic spread, in this paper, we construct several mathematical models (including without delay, with a coupling delay, and with double delays) of epidemic synchronization by applying the adaptive feedback motivated by real observations. By using Lyapunov function methods, we obtain the conditions for local and global stability of these epidemic synchronization models. Then, we illustrate that quenched mean-field theory is more accurate than heterogeneous mean-field theory in the prediction of epidemic synchronization. Finally, some numerical simulations are performed to complement our theoretical results, which also reveal some unexpected phenomena, for example, the coupling delay and epidemic delay influence the speed of epidemic synchronization. This work makes further exploration on the relationship between epidemic dynamics and synchronization dynamics, in the hope of being helpful to the study of other dynamical phenomena in the process of epidemic spread.

Synchronizing Genetic Oscillators by Signaling Molecules

The authors examine collective rhythms in a general multicell system with both linearly diffusive and nondiffusive couplings. The effect of coupling on synchronization through intercellular signaling in a population of Escherichia coli cells is studied. In particular, a synchronization solution is given through the auxiliary individual system for 2 types of couplings. The sufficient conditions for the global synchronization of such a coupled system are derived based on the Lyapunov function method. The authors show that an appropriate design of the coupling and the inner-linking matrix can ensure global synchronization of the coupled synthetic biological system. Moreover, they demonstrate that the dynamics of an individual cell with coupling and without coupling may be qualitatively different; one is oscillatory, and the other is steady state. The change from a nonoscillatory state to an oscillatory one is induced by appropriate coupling, which also entrains all cells to synchronization. These results establish not only a theoretical foundation but also a quantitative basis for understanding the essential cooperative dynamics, such as collective rhythms or synchronization, in a population of cells.

Global synchronization of discrete-time coupled neural networks with Markovian switching and impulses

This paper is concerned with the problem of synchronization analysis for discrete-time coupled neural networks. The networks under consideration are subject to: (1) the jumping parameters that are modeled as a continuous-time, discrete-state Markov process; (2) impulsive disturbances; and (3) time delays that include both the mode-dependent discrete and distributed delay. By constructing suitable Lyapunov–Krasovskii functional and combining with linear matrix inequality approach, several novel criteria are derived for verifying the global exponential synchronization in the mean square of such stochastic dynamical networks. The derived conditions are established in terms of linear matrix inequalities, which can be easily solved by some available software packages. A simulation example is presented to show the effectiveness and applicability of the obtained results.

Velocity synchronization of multi-agent systems with mismatched parameters via sampled position data

Power systems are special multi-agent systems with nonlinear coupling function and symmetric structures. This paper extends these systems to a class of multi-agent systems with mismatched parameters, linear coupling function, and asymmetric structures and investigates their velocity synchronization via sampled position data. The dynamics of the agents is adopted as that of generators with mismatched parameters, while the system structures are supposed to be complex. Two distributed linear consensus protocols are designed, respectively, for multi-agent systems without or with communication delay. Necessary and sufficient conditions based on the sampling period, the mismatched parameters, the delay, and the nonzero eigenvalues of the Laplacian matrix are established. It is shown that velocity synchronization of multi-agent systems with mismatched parameters can be achieved if the sampled period is chosen appropriately. Simulations are given to illustrate the effectiveness of the theoretical results.

Exponential Synchronization of Coupled Stochastic Memristor-Based Neural Networks With Time-Varying Probabilistic Delay Coupling and Impulsive Delay

This paper deals with the exponential synchronization of coupled stochastic memristor-based neural networks with probabilistic time-varying delay coupling and time-varying impulsive delay. There is one probabilistic transmittal delay in the delayed coupling that is translated by a Bernoulli stochastic variable satisfying a conditional probability distribution. The disturbance is described by a Wiener process. Based on Lyapunov functions, Halanay inequality, and linear matrix inequalities, sufficient conditions that depend on the probability distribution of the delay coupling and the impulsive delay were obtained. Numerical simulations are used to show the effectiveness of the theoretical results.