Cluster Synchronization In Networks Research Articles

One-way dependent clusters and stability of cluster synchronization in directed networks

Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.

Analyzing synchronized clusters in neuron networks

The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. Common approaches to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to two examples with different size and features (the directed network of the macaque cerebral cortex and the swim central pattern generator of a mollusc) provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some modeling simplifications in one of the examples, the obtained results are in good agreement with previously reported biological data.

Symmetries and cluster synchronization in multilayer networks

Real-world systems in epidemiology, social sciences, power transportation, economics and engineering are often described as multilayer networks. Here we first define and compute the symmetries of multilayer networks, and then study the emergence of cluster synchronization in these networks. We distinguish between independent layer symmetries, which occur in one layer and are independent of the other layers, and dependent layer symmetries, which involve nodes in different layers. We study stability of the cluster synchronous solution by decoupling the problem into a number of independent blocks and assessing stability of each block through a Master Stability Function. We see that blocks associated with dependent layer symmetries have a different structure to the other blocks, which affects the stability of clusters associated with these symmetries. Finally, we validate the theory in a fully analog experiment in which seven electronic oscillators of three kinds are connected with two kinds of coupling.

Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators

In this paper we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected network. Cluster synchronization is at the basis of several biological and technological processes; yet the underlying mechanisms to enable cluster synchronization of Kuramoto oscillators have remained elusive. In this paper we derive quantitative conditions on the network weights, cluster configuration, and oscillators' natural frequency that ensure asymptotic stability of the cluster synchronization manifold; that is, the ability to recover the desired cluster synchronization configuration following a perturbation of the oscillators' states. Qualitatively, our results show that cluster synchronization is stable when the intra-cluster coupling is sufficiently stronger than the inter-cluster coupling, the natural frequencies of the oscillators in distinct clusters are sufficiently different, or, in the case of two clusters, when the intra-cluster dynamics is homogeneous. We illustrate and validate the effectiveness of our theoretical results via numerical studies.

Cluster synchronization in networked nonidentical chaotic oscillators.

In exploring oscillator synchronization, a general observation is that as the oscillators become nonidentical, e.g., introducing parameter mismatch among the oscillators, the propensity for synchronization will be deteriorated. Yet in realistic systems, parameter mismatch is unavoidable and even worse in some circumstances, the oscillators might follow different types of dynamics. Considering the significance of synchronization to the functioning of many realistic systems, it is natural to ask the following question: Can synchronization be achieved in networked oscillators of clearly different parameters or dynamics? Here, by the model of networked chaotic oscillators, we are able to demonstrate and argue that, despite the presence of parameter mismatch (or different dynamics), stable synchronization can still be achieved on symmetric complex networks. Specifically, we find that when the oscillators are configured on the network in such a way that the symmetric nodes have similar parameters (or follow the same type of dynamics), cluster synchronization can be generated. The stabilities of the cluster synchronization states are analyzed by the method of symmetry-based stability analysis, with the theoretical predictions in good agreement with the numerical results. Our study sheds light on the interplay between symmetry and cluster synchronization in complex networks and give insights into the functionalities of realistic systems where nonidentical nonlinear oscillators are presented and cluster synchronization is crucial.

A criterion for stability of cluster synchronization in networks with external equitable partitions

Networks of nonlinear dynamical systems may exhibit a collective state, known as cluster synchronization, characterized by the formation of groups, where the units within each cluster converge to the same behavior that is distinct for each cluster. In this brief paper we derive a criterion for local stability in Milnor sense of this synchronous state that is expressed as a simple relationship between the maximum Lyapunov exponent of the uncoupled dynamics, the coupling coefficient, and one eigenvalue of the graph Laplacian spectrum. In the last part of the paper we discuss the application of the criterion to networks design and control.

Cluster synchronization in networks of structured communities

We investigate the synchronization problem for a network of subnetwork with community structure. We consider a model with different levels of interconnection: At the first, strongly coupled adjacent nodes are modeled as a compact unit (CU) where connections are uniformed and undirected. A second level of connection occurs as CUs with similar dynamic behaviors from communities by weighted and undirected interactions. Finally a third level of connection, where directed and weighted connections between these communities form the entire network of subnetworks. Using the Lyapunov approach we show that cluster synchronization is possible for our network model where even though all nodes are interconnected each community achieves complete synchronization within, while the behavior between communities is not synchronized. Our main results are illustrated using numerical simulations of well-known benchmark systems.

Symmetry- and input-cluster synchronization in networks.

We study cluster synchronization in networks and show that the stability of all possible cluster synchronization patterns depends on a small set of Lyapunov exponents. Our approach can be applied to clusters corresponding to both orbital partitions of the network nodes (symmetry-cluster synchronization) and equitable partitions of the network nodes (input-cluster synchronization). Our results are verified experimentally in networks of coupled optoelectronic oscillators.

Clusters in nonsmooth oscillator networks.

For coupled oscillator networks with Laplacian coupling, the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory, this approach has recently been extended to treat more general cluster states. However, the MSF and its generalizations require the determination of a set of Floquet multipliers from variational equations obtained by linearization around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by, the framework is often explored using numerical techniques. Here, we show that further insight into network dynamics can be obtained by focusing on piecewise linear (PWL) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of saltation matrices. By augmenting the variational approach for studying smooth dynamical systems with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration, we analyze an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled network built from planar PWL nodes, including a reduction of the popular Morris-Lecar neuron model. We use these examples to emphasize that the stability of network cluster states can depend as much on the choice of single node dynamics as it does on the form of network structural connectivity. Importantly, the procedure that we present here, for understanding cluster synchronization in networks, is valid for a wide variety of systems in biology, physics, and engineering that can be described by PWL oscillators.

Almost sure cluster synchronization of Markovian switching complex networks with stochastic noise via decentralized adaptive pinning control

This paper investigates the issue of almost sure cluster synchronization in nonlinearly coupled complex networks with nonidentical nodes and time-varying delay. These networks are modulated by a continuous-time Markov chain and disturbed by a Brownian movement. The decentralized adaptive update law and pinning control protocol are employed in designing controllers for guaranteeing almost sure cluster synchronization. By constructing a novel stochastic Lyapunov–Krasovskii function and using the stochastic Lasalle-type invariance theorem, some sufficient conditions for almost sure cluster synchronization of the networks are derived. Finally, a numerical example is given to testify the effectiveness of the theoretical results.

Approximate cluster synchronization in networks with symmetries and parameter mismatches.

We study cluster synchronization in networks with symmetries in the presence of small generic parametric mismatches of two different types: mismatches affecting the dynamics of the individual uncoupled systems and mismatches affecting the network couplings. We perform a stability analysis of the nearly synchronous cluster synchronization solution and reduce the stability problem to a low-dimensional form. We also show how under certain conditions the low dimensional analysis can be used to predict the overall synchronization error, i.e., how close the individual nearly synchronous trajectories are to each other.

Graph partitions and cluster synchronization in networks of oscillators.

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.

Adaptive pinning control of cluster synchronization in complex networks with Lurie-type nonlinear dynamics

In this paper, cluster synchronization problem is investigated for a kind of complex dynamical networks with Lurie-type non-linear dynamics. With the aid of a local update law, a pinning adaptive control strategy is proposed to solve the cluster synchronization problem of the networks. Some simple criteria for cluster synchronization are presented for the networks with external disturbances and time delays. Firstly, sufficient conditions are established to realize cluster synchronization of the networks without disturbances and delays. Then, the cluster synchronization problem of the networks with external disturbances is considered. Finally, the cluster synchronization in the networks with time-varying delays is further studied. Numerical examples are given to verify and illustrate the theoretical results.

Cluster synchronization in networks of neurons with chemical synapses

In this work, we study the cluster synchronization of chemically coupled and generally formulated networks which are allowed to be nonidentical. The sufficient condition for the existence of stably synchronous clusters is derived. Specifically, we only need to check the stability of the origins of m decoupled linear systems. Here, m is the number of subpopulations. Examples of nonidentical networks such as Hindmarsh-Rose (HR) neurons with various choices of parameters in different subpopulations, or HR neurons in one subpopulation and FitzHugh-Nagumo neurons in the other subpopulation are provided. Explicit threshold for the coupling strength that guarantees the stably cluster synchronization can be obtained.

Clustering in delay-coupled smooth and relaxational chemical oscillators

We investigate cluster synchronization in networks of nonlinear systems with time-delayed coupling. Using a generic model for a system close to the Hopf bifurcation, we predict the order of appearance of different cluster states and their corresponding common frequencies depending upon coupling delay. We may tune the delay time in order to ensure the existence and stability of a specific cluster state. We qualitatively and quantitatively confirm these results in experiments with chemical oscillators. The experiments also exhibit strongly nonlinear relaxation oscillations as we increase the voltage, i.e., go further away from the Hopf bifurcation. In this regime, we find secondary cluster states with delay-dependent phase lags. These cluster states appear in addition to primary states with delay-independent phase lags observed near the Hopf bifurcation. Extending the theory on Hopf normal-form oscillators, we are able to account for realistic interaction functions, yielding good agreement with experimental findings.

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.

Dynamics, control and information in delay-coupled systems: an overview

Dynamics, control and information in delay-coupled systems: an overview

Adaptive cluster synchronization in networks with time-varying and distributed coupling delays

This paper studies the adaptive cluster synchronization of a generalized linearly coupled network with time-varying delay and distributed delays. This network includes nonidentical nodes displaying different local dynamical behaviors, while for each cluster of that network the internal dynamics is uniform (such as chaotic, periodic, or stable behavior). In particular, the generalized coupling matrix of this network can be asymmetric and weighted. Two different adaptive laws of time-varying coupling strength and a linear feedback control are designed to achieve the cluster synchronization of this network. Some sufficient conditions to ensure the cluster synchronization are obtained by using the invariant principle of functional differential equations and linear matrix inequality (LMI). Numerical simulations verify the efficiency of our proposed adaptive control method.

Cluster synchronization in networks of partial-state coupled linear systems via pinning control technique

This paper investigates the cluster synchronization for network of linear systems under general topology via pinning control technique. It is shown that an appropriate feedback controller can be designed to achieve the given cluster synchronization pattern if the induced network topology of each cluster has a directed spanning tree and further compared to the couplings among different clusters, the couplings within the each cluster are sufficiently strong. Both the lower bounds for such strengths of couplings and the cluster synchronization rate are specified as well. Finally, some illustrative examples are provided to demonstrate the effectiveness of the theoretical findings.

Cluster synchronization in networks of distinct groups of maps

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.