Synchronization in Networks of Linearly Coupled Dynamical Systems via Event-Triggered Diffusions

In this paper, we utilize event-triggered coupling configurations to realize synchronization of linearly coupled dynamical systems. Here, the diffusion couplings are set up from the latest observations of the nodes and their neighborhood and the next observation time is triggered by the proposed criteria based on the local neighborhood information as well. Two scenarios are considered: 1) continuous monitoring, in which each node can observe its neighborhood's instantaneous states and 2) discrete monitoring, in which each node can obtain only its neighborhood's states at the same time point when the coupling term is triggered. In both the cases, we prove that if the system with persistent coupling can synchronize, then these event-triggered coupling strategies can synchronize the system too.

Synchronization of Memristor-Based Coupling Recurrent Neural Networks With Time-Varying Delays and Impulses

Synchronization of an array of linearly coupled memristor-based recurrent neural networks with impulses and time-varying delays is investigated in this brief. Based on the Lyapunov function method, an extended Halanay differential inequality and a new delay impulsive differential inequality, some sufficient conditions are derived, which depend on impulsive and coupling delays to guarantee the exponential synchronization of the memristor-based recurrent neural networks. Impulses with and without delay and time-varying delay are considered for modeling the coupled neural networks simultaneously, which renders more practical significance of our current research. Finally, numerical simulations are given to verify the effectiveness of the theoretical results.

Driving-based generalized synchronization in two-layer networks via pinning control

Synchronization of complex networks has been extensively investigated in various fields. In the real world, one network is usually affected by another one but coexists in harmony with it, which can be regarded as another kind of synchronization--generalized synchronization (GS). In this paper, the GS in two-layer complex networks with unidirectional inter-layer coupling via pinning control is investigated based on the auxiliary-system approach. Specifically, for two-layer networks under study, one is considered as the drive network and the other is the response one. According to the auxiliary-system approach, output from the drive layer is designed as input for the response one, and an identical duplication of the response layer is constructed, which is driven by the same driving signals. A sufficient condition for achieving GS via pinning control is presented. Numerical simulations are further provided to illustrate the correctness of the theoretical results. It is also revealed that the least number of pinned nodes needed for achieving GS decreases with the increasing density of the response layer. In addition, it is found that when the intra-layer coupling strength of the response network is large, nodes with larger degrees should be selected to pin first for the purpose of achieving GS. However, when the coupling strength is small, it is more preferable to pin nodes with smaller degrees. This work provides engineers with a convenient approach to realize harmonious coexistence of various complex systems, which can further facilitate the selection of pinned systems and reduce control cost.

Mode-Dependent Stochastic Synchronization for Markovian Coupled Neural Networks With Time-Varying Mode-Delays

This paper investigates the stochastic synchronization problem for Markovian hybrid coupled neural networks with interval time-varying mode-delays and random coupling strengths. The coupling strengths are mutually independent random variables and the coupling configuration matrices are nonsymmetric. A mode-dependent augmented Lyapunov-Krasovskii functional (LKF) is proposed, where some terms involving triple or quadruple integrals are considered, which makes the LKF matrices mode-dependent as much as possible. This gives significant improvement in the synchronization criteria, i.e., less conservative results can be obtained. In addition, by applying an extended Jensen's integral inequality and the properties of random variables, new delay-dependent synchronization criteria are derived. The obtained criteria depend not only on upper and lower bounds of mode-delays but also on mathematical expectations and variances of the random coupling strengths. Finally, two numerical examples are provided to demonstrate the feasibility of the proposed results.

Synchronization of linearly coupled networks with delays via aperiodically intermittent pinning control

In this paper, we investigate the exponential synchronization problem for linearly coupled networks with delay by pinning a simple aperiodically intermittent controller. The network topology can be directed. Different from previous works, the intermittent control can be aperiodic. Two types of delay are considered. The first case is that the delay is time-varying and large, and in this case, there is no restriction imposed on the delay and the control (and/or rest) width. The other one is that the delay is small enough so that it is less than the minimum of control width. Different approaches are provided to investigate these two cases, and some criteria are given to realize exponential synchronization. Furthermore, by applying the adaptive approach to the second model, we establish a general adaptive theory for intermittent control, which can be applied not only to networks without time delay, but also to delayed networks, regardless of whether the intermittent control is periodic or aperiodic. Finally, the numerical simulations are given to verify the validness of the theoretical results.

Power-rate synchronization of coupled genetic oscillators with unbounded time-varying delay

In this paper, a new synchronization problem for the collective dynamics among genetic oscillators with unbounded time-varying delay is investigated. The dynamical system under consideration consists of an array of linearly coupled identical genetic oscillators with each oscillators having unbounded time-delays. A new concept called power-rate synchronization, which is different from both the asymptotical synchronization and the exponential synchronization, is put forward to facilitate handling the unbounded time-varying delays. By using a combination of the Lyapunov functional method, matrix inequality techniques and properties of Kronecker product, we derive several sufficient conditions that ensure the coupled genetic oscillators to be power-rate synchronized. The criteria obtained in this paper are in the form of matrix inequalities. Illustrative example is presented to show the effectiveness of the obtained results.

An Integrated Approach to Global Synchronization and State Estimation for Nonlinear Singularly Perturbed Complex Networks

This paper aims to establish a unified framework to handle both the exponential synchronization and state estimation problems for a class of nonlinear singularly perturbed complex networks (SPCNs). Each node in the SPCN comprises both "slow" and "fast" dynamics that reflects the singular perturbation behavior. General sector-like nonlinear function is employed to describe the nonlinearities existing in the network. All nodes in the SPCN have the same structures and properties. By utilizing a novel Lyapunov functional and the Kronecker product, it is shown that the addressed SPCN is synchronized if certain matrix inequalities are feasible. The state estimation problem is then studied for the same complex network, where the purpose is to design a state estimator to estimate the network states through available output measurements such that dynamics (both slow and fast) of the estimation error is guaranteed to be globally asymptotically stable. Again, a matrix inequality approach is developed for the state estimation problem. Two numerical examples are presented to verify the effectiveness and merits of the proposed synchronization scheme and state estimation formulation. It is worth mentioning that our main results are still valid even if the slow subsystems within the network are unstable.

Global exponential synchronization of multiple memristive neural networks with time delay via nonlinear coupling

This paper presents theoretical results on the global exponential synchronization of multiple memristive neural networks with time delays. A novel coupling scheme is introduced, in a general topological structure described by a directed or undirected graph, with a linear diffusive term and discontinuous sign term. Several criteria are derived based on the Lyapunov stability theory to ascertain the global exponential stability of synchronization manifold in the coupling scheme. Simulation results for several examples are given to substantiate the effectiveness of the theoretical results.

Network synchronization of time-delayed coupled nonlinear systems using predictor-based diffusive dynamic couplings

We study the problem of controlled network synchronization of coupled semipassive systems in the case when the outputs (the coupling variables) and the inputs are subject to constant time-delay (as it is often the case in a networked context). Predictor-based dynamic output feedback controllers are proposed to interconnect the systems on a given network. Using Lyapunov-Krasovskii functional and the notion of semipassivity, we prove that under some mild assumptions, the solutions of the interconnected systems are globally ultimately bounded. Sufficient conditions on the systems to be interconnected, on the network topology, on the coupling dynamics, and on the time-delays that guarantee global state synchronization are derived. A local analysis is provided in which we compare the performance of our predictor-based control scheme against the existing static diffusive couplings available in the literature. We show (locally) that the time-delay that can be induced to the network may be increased by including the predictors in the loop. The results are illustrated by computer simulations of coupled Hindmarsh-Rose neurons.

Synchronization of nonlinear coupled networks via aperiodically intermittent pinning control

In this paper, pinning synchronization problem for nonlinear coupled networks is investigated, which can be recurrently connected neural networks, cellular neural networks, Hodgkin-Huxley models, Lorenz chaotic oscillators, and so on. Nodes in the network are assumed to be identical and nodes' dynamical behaviors are described by continuous-time equations. The network topology is undirected and static. At first, the scope of accepted nonlinear coupling functions is defined, and the effect of nonlinear coupling functions on synchronization is carefully discussed. Then, the pinning control technique is used for synchronization, especially the control type is aperiodically intermittent. Some sufficient conditions to guarantee global synchronization are presented. Furthermore, the adaptive approach is also applied on the pinning control, and a centralized adaptive algorithm is designed and its validity is also proved. Finally, several numerical simulations are given to verify the obtained theoretical results.

Synchronization on complex networks of networks

In this paper, pinning synchronization on complex networks of networks is investigated, where there are many subnetworks with the interactions among them. The subnetworks and their connections can be regarded as the nodes and interactions of the networks, respectively, which form the networks of networks. In this new setting, the aim is to design pinning controllers on the chosen nodes of each subnetwork so as to reach synchronization behavior. Some synchronization criteria are established for reaching pinning control on networks of networks. Furthermore, the pinning scheme is designed, which shows that the nodes with very low degrees and large degrees are good candidates for applying pinning controllers. Then, the attack and robustness of the pinning scheme are discussed. Finally, a simulation example is presented to verify the theoretical analysis in this paper.

H ∞ cluster synchronization for a class of neutral complex dynamical networks with Markovian switching

H_∞ cluster synchronization problem for a class of neutral complex dynamical networks (NCDNs) with Markovian switching is investigated in this paper. Both the retarded and neutral delays are considered to be interval mode dependent and time varying. The concept of H_∞ cluster synchronization is proposed to quantify the attenuation level of synchronization error dynamics against the exogenous disturbance of the NCDNs. Based on a novel Lyapunov functional, by employing some integral inequalities and the nature of convex combination, mode delay-range-dependent H_∞ cluster synchronization criteria are derived in the form of linear matrix inequalities which depend not only on the disturbance attenuation but also on the initial values of the NCDNs. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the proposed theoretical results.

Pinning impulsive control algorithms for complex network

In this paper, we further investigate the synchronization of complex dynamical network via pinning control in which a selection of nodes are controlled at discrete times. Different from most existing work, the pinning control algorithms utilize only the impulsive signals at discrete time instants, which may greatly improve the communication channel efficiency and reduce control cost. Two classes of algorithms are designed, one for strongly connected complex network and another for non-strongly connected complex network. It is suggested that in the strongly connected network with suitable coupling strength, a single controller at any one of the network's nodes can always pin the network to its homogeneous solution. In the non-strongly connected case, the location and minimum number of nodes needed to pin the network are determined by the Frobenius normal form of the coupling matrix. In addition, the coupling matrix is not necessarily symmetric or irreducible. Illustrative examples are then given to validate the proposed pinning impulsive control algorithms.

Impulsive control for synchronizing delayed discrete complex networks with switching topology

In this paper, global exponential synchronization of a class of discrete delayed complex networks with switching topology has been investigated by using Lyapunov-Ruzimiki method. The impulsive scheme is designed to work at the time instant of switching occurrence. A time-varying delay-dependent criterion for impulsive synchronization is given to ensure the delayed discrete complex networks switching topology tending to a synchronous state. Furthermore, a numerical simulation is given to illustrate the effectiveness of main results

Swarming behaviors in multi-agent systems with nonlinear dynamics

The dynamic analysis of a continuous-time multi-agent swarm model with nonlinear profiles is investigated in this paper. It is shown that, under mild conditions, all agents in a swarm can reach cohesion within a finite time, where the upper bounds of the cohesion are derived in terms of the parameters of the swarm model. The results are then generalized by considering stochastic noise and switching between nonlinear profiles. Furthermore, swarm models with limited sensing range inducing changing communication topologies and unbounded repulsive interactions between agents are studied by switching system and nonsmooth analysis. Here, the sensing range of each agent is limited and the possibility of collision among nearby agents is high. Finally, simulation results are presented to demonstrate the validity of the theoretical analysis.

Cluster synchronization induced by one-node clusters in networks with asymmetric negative couplings

This paper deals with the problem of cluster synchronization in networks with asymmetric negative couplings. By decomposing the coupling matrix into three matrices, and employing Lyapunov function method, sufficient conditions are derived for cluster synchronization. The conditions show that the couplings of multi-node clusters from one-node clusters have beneficial effects on cluster synchronization. Based on the effects of the one-node clusters, an effective and universal control scheme is put forward for the first time. The obtained results may help us better understand the relation between cluster synchronization and cluster structures of the networks. The validity of the control scheme is confirmed through two numerical simulations, in a network with no cluster structure and in a scale-free network.

Oscillations and synchrony in a cortical neural network

In this paper, the oscillations and synchronization status of two different network connectivity patterns based on Izhikevich model are studied. One of the connectivity patterns is a randomly connected neuronal network, the other one is a small-world neuronal network. This Izhikevich model is a simple model which can not only reproduce the rich behaviors of biological neurons but also has only two equations and one nonlinear term. Detailed investigations reveal that by varying some key parameters, such as the connection weights of neurons, the external current injection, the noise of intensity and the neuron number, this neuronal network will exhibit various collective behaviors in randomly coupled neuronal network. In addition, we show that by changing the number of nearest neighbor and connection probability in small-world topology can also affect the collective dynamics of neuronal activity. These results may be instructive in understanding the collective dynamics of mammalian cortex.

Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects

This paper investigates the synchronization problem of coupled switched neural networks (SNNs) with mode-dependent impulsive effects and time delays. The main feature of mode-dependent impulsive effects is that impulsive effects can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching. The impulses considered here include those that suppress synchronization or enhance synchronization. Based on switching analysis techniques and the comparison principle, the exponential synchronization criteria are derived for coupled delayed SNNs with mode-dependent impulsive effects. Finally, simulations are provided to illustrate the effectiveness of the results.

Robust global synchronization of two complex dynamical networks

We investigate the synchronization of two coupled complex dynamical networks, a problem that has been termed outer synchronization in the literature. Our approach relies on (a) a basic lemma on the eigendecomposition of matrices resulting from Kronecker products and (b) a suitable choice of Lyapunov function related to the synchronization error dynamics. Starting from these two ingredients, a theorem that provides a sufficient condition for outer synchronization of the networks is proved. The condition in the theorem is expressed as a linear matrix inequality. When satisfied, synchronization is guaranteed to occur globally, i.e., independently of the initial conditions of the networks. The argument of the proof includes the design of the gain of the synchronizer, which is a constant square matrix with dimension dependent on the number of dynamic variables in a single network node, but independent of the size of the overall network, which can be much larger. This basic result is subsequently elaborated to simplify the design of the synchronizer, to avoid unnecessarily restrictive assumptions (e.g., diffusivity) on the coupling matrix that defines the topology of the networks and, finally, to obtain synchronizers that are robust to model errors in the parameters of the coupled networks. An illustrative numerical example for the outer synchronization of two networks of classical Lorenz nodes with perturbed parameters is presented.

Distributed synchronization of coupled neural networks via randomly occurring control

In this paper, we study the distributed synchronization and pinning distributed synchronization of stochastic coupled neural networks via randomly occurring control. Two Bernoulli stochastic variables are used to describe the occurrences of distributed adaptive control and updating law according to certain probabilities. Both distributed adaptive control and updating law for each vertex in a network depend on state information on each vertex's neighborhood. By constructing appropriate Lyapunov functions and employing stochastic analysis techniques, we prove that the distributed synchronization and the distributed pinning synchronization of stochastic complex networks can be achieved in mean square. Additionally, randomly occurring distributed control is compared with periodically intermittent control. It is revealed that, although randomly occurring control is an intermediate method among the three types of control in terms of control costs and convergence rates, it has fewer restrictions to implement and can be more easily applied in practice than periodically intermittent control.

Distributed Synchronization in Networks of Agent Systems With Nonlinearities and Random Switchings

In this paper, the distributed synchronization problem of networks of agent systems with controllers and nonlinearities subject to Bernoulli switchings is investigated. Controllers and adaptive updating laws injected in each vertex of networks depend on the state information of its neighborhood. Three sets of Bernoulli stochastic variables are introduced to describe the occurrence probabilities of distributed adaptive controllers, updating laws and nonlinearities, respectively. By the Lyapunov functions method, we show that the distributed synchronization of networks composed of agent systems with multiple randomly occurring nonlinearities, multiple randomly occurring controllers, and multiple randomly occurring updating laws can be achieved in mean square under certain criteria. The conditions derived in this paper can be solved by semi-definite programming. Moreover, by mathematical analysis, we find that the coupling strength, the probabilities of the Bernoulli stochastic variables, and the form of nonlinearities have great impacts on the convergence speed and the terminal control strength. The synchronization criteria and the observed phenomena are demonstrated by several numerical simulation examples. In addition, the advantage of distributed adaptive controllers over conventional adaptive controllers is illustrated.

Partial synchronization in diffusively time-delay coupled oscillator networks

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.

Neurocircuitry for modeling drug effects

The identification and functional understanding of the neurocircuitry that mediates alcohol and drug effects that are relevant for the development of addictive behavior is a fundamental challenge in addiction research. Here we introduce an assumption-free construction of a neurocircuitry that mediates acute and chronic drug effects on neurotransmitter dynamics that is solely based on rodent neuroanatomy. Two types of data were considered for constructing the neurocircuitry: (1) information on the cytoarchitecture and neurochemical connectivity of each brain region of interest obtained from different neuroanatomical techniques; (2) information on the functional relevance of each region of interest with respect to alcohol and drug effects. We used mathematical data mining and hierarchical clustering methods to achieve the highest standards in the preprocessing of these data. Using this approach, a dynamical network of high molecular and spatial resolution containing 19 brain regions and seven neurotransmitter systems was obtained. Further graph theoretical analysis suggests that the neurocircuitry is connected and cannot be separated into further components. Our analysis also reveals the existence of a principal core subcircuit comprised of nine brain regions: the prefrontal cortex, insular cortex, nucleus accumbens, hypothalamus, amygdala, thalamus, substantia nigra, ventral tegmental area and raphe nuclei. Finally, by means of algebraic criteria for synchronizability of the neurocircuitry, the suitability for in silico modeling of acute and chronic drug effects is indicated. Indeed, we introduced as an example a dynamical system for modeling the effects of acute ethanol administration in rats and obtained an increase in dopamine release in the nucleus accumbens-a hallmark of drug reinforcement-to an extent similar to that seen in numerous microdialysis studies. We conclude that the present neurocircuitry provides a structural and dynamical framework for large-scale mathematical models and will help to predict chronic drug effects on brain function.

Adaptive node-to-node pinning synchronization control of complex networks

In this work, we propose an adaptive node-to-node pinning control strategy. In this approach, both the coupling strength among nodes and the pinning control gains are adaptively changed according to well chosen adaptation laws that take into account the specificities of the oscillators and the network topology. Proof of stability and performance comparison is also shown in this paper.

Partial synchronization in stochastic dynamical networks with switching communication channels

In this paper, the partial synchronization problem of stochastic dynamical networks (SDNs) is investigated. Unlike the existing models, the SDN considered in this paper suffers from a class of communication constraint-only part of nodes' states can be transmitted. Thus, less nodes' states can be used to synchronize the SDN, which makes the analysis of the synchronization problem much harder. A set of channel matrices are introduced to reflect such kind of constraint. Furthermore, due to unpredictable environmental changes, the channel matrices can switch among some communication modes. The switching considered here is governed by a Markov process. To overcome the difficulty, a regrouping method is employed to derive our main results. The obtained conditions guarantee that partial synchronization can be achieved for SDNs under switching communication constraint. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results and how the communication constraint influences synchronization result.

Impulsive synchronization of coupled dynamical networks with nonidentical Duffing oscillators and coupling delays

This paper aims to investigate the synchronization problem of coupled dynamical networks with nonidentical Duffing-type oscillators without or with coupling delays. Different from cluster synchronization of nonidentical dynamical networks in the previous literature, this paper focuses on the problem of complete synchronization, which is more challenging than cluster synchronization. By applying an impulsive controller, some sufficient criteria are obtained for complete synchronization of the coupled dynamical networks of nonidentical oscillators. Furthermore, numerical simulations are given to verify the theoretical results.

Exponential cluster synchronization of impulsive delayed genetic oscillators with external disturbances

This paper investigates the problem of the exponential cluster synchronization of coupled impulsive genetic oscillators with external disturbances and communication delay. Based on the Kronecker product, some new cluster synchronization criteria for coupled impulsive genetic oscillators with attenuation level are derived. The derived results are related to the impulsive strength, and the derived results also indicate that the maximal allowable bound of time delay is inversely proportional to the decay rate, the decay rate is proportional to the couple strength, the maximal allowable bound of time delay is proportional to attenuation level, and the attenuation level is inversely proportional to the couple strength. Moreover, the case when the feedback have different self-delay is also investigated. Finally, numerical examples are given to illustrate the effectiveness of the derived results.

Synchronization of impulsively coupled complex systems with delay

This paper investigates the synchronization of complex systems with delay that are impulsively coupled at discrete instants only. Based on the comparison theorem of impulsive differential system, a distributed impulsive control scheme is proposed to achieve the synchronization for systems with delay. In the control strategy, the influence of all nodes to network synchronization relies on its weight. The proposed control scheme is applied to the chaotic delayed Hopfield neural networks and numerical simulations are presented to demonstrate the effectiveness of the proposed scheme.

Adaptive mechanism between dynamical synchronization and epidemic behavior on complex networks

Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks.

Robust outer synchronization between two complex networks with fractional order dynamics

Synchronization between two coupled complex networks with fractional-order dynamics, hereafter referred to as outer synchronization, is investigated in this work. In particular, we consider two systems consisting of interconnected nodes. The state variables of each node evolve with time according to a set of (possibly nonlinear and chaotic) fractional-order differential equations. One of the networks plays the role of a master system and drives the second network by way of an open-plus-closed-loop (OPCL) scheme. Starting from a simple analysis of the synchronization error and a basic lemma on the eigenvalues of matrices resulting from Kronecker products, we establish various sets of conditions for outer synchronization, i.e., for ensuring that the errors between the state variables of the master and response systems can asymptotically vanish with time. Then, we address the problem of robust outer synchronization, i.e., how to guarantee that the states of the nodes converge to common values when the parameters of the master and response networks are not identical, but present some perturbations. Assuming that these perturbations are bounded, we also find conditions for outer synchronization, this time given in terms of sets of linear matrix inequalities (LMIs). Most of the analytical results in this paper are valid both for fractional-order and integer-order dynamics. The assumptions on the inner (coupling) structure of the networks are mild, involving, at most, symmetry and diffusivity. The analytical results are complemented with numerical examples. In particular, we show examples of generalized and robust outer synchronization for networks whose nodes are governed by fractional-order Lorenz dynamics.