Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.     

Shuttle-run synchronization in mobile ad hoc networks

In this work, we study the collective dynamics of phase oscillators in a mobile ad hoc network whose topology changes dynamically. As the network size or the communication radius of individual oscillators increases, the topology of the ad hoc network first undergoes percolation, forming a giant cluster, and then gradually achieves global connectivity. It is shown that oscillator mobility generally enhances the coherence in such networks. Interestingly, we find a new type of phase synchronization/clustering, in which the phases of the oscillators are distributed in a certain narrow range, while the instantaneous frequencies change signs frequently, leading to shuttle-run-like motion of the oscillators in phase space. We conduct a theoretical analysis to explain the mechanism of this synchronization and obtain the critical transition point.     

Synchronization in networks of limit cycle oscillators

We investigate the synchronization behaviour of three different networks of nonlinearly coupled oscillators. Each network consists of several clusters of oscillators, and the clusters themselves consist of any number of oscillators. In each cluster the eigenfrequencies scatter around the cluster frequency (mean frequency). The coupling strength varies in each cluster, too. We analyze the synchronized states by means of the center manifold theorem. This enables us to calculate these states explicitly, and to prove their stability. Moreover we are able to determine frequency shifts caused by different coupling mechanisms. In a number of cases we calculate the synchronisation threshold explicitely. Numerical simulations illustrate our analytical results. In one of the three networks we have additionally analyzed a single cluster consisting of infinitely many oscillators, that is an oscillatory field. Again, the center manifold theorem enabled us to calculate the synchronized state explicitly and to prove its stability. Our results concerning the oscillatory field are in contradiction to Ermentrout's analysis [6].     

Projective cluster synchronization in drive-response dynamical networks

In this paper, we study the projective cluster synchronization in a drive-response dynamical network with 1+N coupled partially linear chaotic systems. Because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable, pinning control ideas are adopted to direct the different scaling factors onto the desired values. It is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results.     

Synchronization and structure in an adaptive oscillator network

We analyze the interplay of synchronization and structure evolution in an evolving network of phase oscillators. An initially random network is adaptively rewired according to the dynamical coherence of the oscillators, in order to enhance their mutual synchronization. We show that the evolving network reaches a small-world structure. Its clustering coefficient attains a maximum for an intermediate intensity of the coupling between oscillators, where a rich diversity of synchronized oscillator groups is observed. In the stationary state, these synchronized groups are directly associated with network clusters.     

Synchronization transition of identical phase oscillators in a directed small-world network

We numerically study a directed small-world network consisting of attractively coupled, identical phase oscillators. While complete synchronization is always stable, it is not always reachable from random initial conditions. Depending on the shortcut density and on the asymmetry of the phase coupling function, there exists a regime of persistent chaotic dynamics. By increasing the density of shortcuts or decreasing the asymmetry of the phase coupling function, we observe a discontinuous transition in the ability of the system to synchronize. Using a control technique, we identify the bifurcation scenario of the order parameter. We also discuss the relation between dynamics and topology and remark on the similarity of the synchronization transition to directed percolation.     

Frequency clustering of coupled phase oscillators on small-world networks

We analyze the phenomenon of frequency clustering in a system of coupled phase oscillators. The oscillators, which in the absence of coupling have uniformly distributed natural frequencies, are coupled through a small-world network, built according to the Watts-Strogatz model. We study the time evolution and determine variations in the transient times depending on the disorder of the network and on the coupling strength. We investigate the effects of fluctuations in the average frequencies, and discuss the definition of the threshold for synchronization. We characterize the structure of clusters and the distribution of cluster sizes in the synchronization transition, and define suitable order parameters to describe the aggregation of the oscillators as the network disorder and the coupling strength change. The non-monotonic behavior observed in some order parameters is related to fluctuations in the mean frequencies.     

Synchronization in the network of chaotic microwave oscillators

Time scale synchronization in networks of chaotic microwave oscillators with the different topologies of the links between nodes has been studied. As a node element of the network the one-dimensional distributed model of the low-voltage vircator has been used. To characterize the degree of synchronization in the whole network the synchronization index has been introduced. The transition to the synchronous regime is shown to take place via cluster time scale synchronization. Meanwhile, the spectral structure of the output signals is complicated sufficiently which allows using such devices in a number of practical applications.     

Clustering and Synchronization in an Array of Repulsively Coupled Phase Oscillators

Clustering and synchronization in an array of repulsively coupled phase oscillators are numerically investigated. It is found that oscillators are divided into several clusters according to the symmetry in the structure.Synchronization occurs between oscillators in each cluster, while those oscillators belonging to different clusters remain asynchronous. Such synchronization may collapse for all clusters when the dynamics of only one oscillator is altered properly. The synchronous state may return back after a short period of transient process. This is determined by the strength of the oscillator altered. Its application in the communication of one-to-several is suggested.     

Clustering and Synchronization in an Array of Repulsively Coupled Phase Oscillators

Clustering and synchronization in an array of repulsively coupled phase oscillators are numerically investigated. It is found that oscillators are divided into several clusters according to the symmetry in the structure. Synchronization occurs between oscillators in each cluster, while those oscillators belonging to different clusters remain asynchronous. Such synchronization may collapse for all clusters when the dynamics of only one oscillator is altered properly. The synchronous state may return back after a short period of transient process. This is determined by the strength of the oscillator altered. Its application in the communication of one-to-several is suggested.     

Synchronization in lattice-embedded scale-free networks

In this work, we study the effects of embedding a system of non-linear phase oscillators in a two-dimensional scale-free lattice. In order to analyze the effects of the embedding, we consider two different topologies. On the one hand, we consider a scale-free complex network where no constraint on the length of the links is taken into account. On the other hand, we use a method recently introduced for embedding scale-free networks in regular Euclidean lattices. In this case, the embedding is driven by a natural constraint of minimization of the total length of the links in the system. We analyze and compare the synchronization properties of a system of non-linear Kuramoto phase oscillators, when interactions between the oscillators take place in these networks. First, we analyze the behavior of the Kuramoto order parameter and show that the onset of synchronization is lower for non-constrained lattices. Then, we consider the behavior of the mean frequency of the oscillators as a function of the natural frequency for the two different networks and also for different values of the scale-free exponent. We show that, in contrast to non-embedded lattices that present a mean-field-like behavior characterized by the presence of a single cluster of synchronized oscillators, in embedded lattices the presence of a diversity of synchronized clusters at different mean frequencies can be observed. Finally, by considering the behavior of the mean frequency as a function of the degree, we study the role of hubs in the synchronization properties of the system.     

Rich-club network topology to minimize synchronization cost due to phase difference among frequency-synchronized oscillators

As exemplified by power grids and large-scale brain networks, some functions of networks consisting of phase oscillators rely on not only frequency synchronization, but also phase synchronization among the oscillators. Nevertheless, even after the oscillators reach frequency-synchronized status, the phase synchronization is not always accomplished because the phase difference among the oscillators is often trapped at non-zero constant values. Such phase difference potentially results in inefficient transfer of power or information among the oscillators, and avoids proper and efficient functioning of the networks. In the present study, we newly define synchronization cost by using the phase difference among the frequency-synchronized oscillators, and investigate the optimal network structure with the minimum synchronization cost through rewiring-based optimization. By using the Kuramoto model, we demonstrate that the cost is minimized in a network with a rich-club topology, which comprises the densely-connected center nodes and low-degree peripheral nodes connecting with the center module. We also show that the network topology is characterized by its bimodal degree distribution, which is quantified by Wolfson’s polarization index.     

Hierarchical synchronization in complex networks with heterogeneous degrees

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.     

Explosive synchronization of multi-layer frequency-weighted coupled complex systems

Synchronization is a phenomenon that is ubiquitous in engineering and natural ecosystems.The study of explosive synchronization on a single-layer network gives the critical transition coupling strength that causes explosive synchronization.However, no significant findings have been made on multi-layer complex networks.This paper proposes a frequency-weighted Kuramoto model on a two-layer network and the critical coupling strength of explosive synchronization is obtained by both theoretical analysis and numerical validation.It is found that the critical value is affected by the interaction strength between layers and the number of network oscillators.The explosive synchronization will be hindered by enhancing the interaction and promoted by increasing the number of network oscillators.Our results have importance across a range of engineering and biological research fields.     

New Approach to Cluster Synchronization in Complex Dynamical Networks

In this paper, a distributed control strategy is proposed to make a complex dynamical network achieve cluster synchronization, which means that nodes in the same group achieve the same synchronization state, while nodes in different groups achieve different synchronization states. The local and global stability of the cluster synchronization state are analyzed. Moreover, simulation results verify the effectiveness of the new approach.     

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.     

Explosive synchronization in network of mobile oscillators

Recently, the explosive synchronization (ES) has attracted great interests. Motivated by the recent dynamic framework of complex network, we focus on the network of mobile oscillators and study synchronization phenomenon. The local synchronous order parameter of the neighbors of the oscillator is used as the controllable variable to adjust the coupling strength of the oscillator. Hence, it can be seen as a kind of adaptive strategy. By numerical simulation, we find that ES can be observed in the dynamic network of mobile oscillators, accompanying with hysteresis loop, as the coupling strength increases gradually. It is found that the critical value of coupling strength and hysteresis loop width is affected by the natural frequency distribution and the number of neighbors the oscillator owning. It can be deduced that ES will be motivated by increasing the number of oscillators in the network. Meanwhile, our results are feasible to different natural frequency distributions, such as Lorentzian, Gaussian power-law, and Rayleigh distribution, whether it is symmetric or not.     

Explosive synchronization through dynamical environment

We report the emergence of an explosive synchronization transition in the identical oscillators interacting indirectly through a network of dynamical agents. The transition from incoherent state to coherent state and vice–versa in these coupled oscillator exhibits an abrupt as well as irreversible. Such transition depends on the network topology as well as the interaction between the oscillators and dynamical agents rather than degree-frequency correlation in the network of oscillators. The occurrence of explosive synchronization is studied in details by using an appropriate order parameter for limit-cycle oscillators with respect to the different parameters like rewiring probability, average degree, and diffusion rate in dynamical agents.     

Synchronization speed of identical oscillators on community networks

Synchronizability of complex oscillators networks has attracted much research interest in recent years. In contrast, in this paper we investigate numerically the synchronization speed, rather than the synchronizability or synchronization stability, of identical oscillators on complex networks with communities. A new weighted community network model is employed here, in which the community strength could be tunable by one parameter δ. The results showed that the synchronization speed of identical oscillators on community networks could reach a maximal value when δ is around 0.1. We argue that this is induced by the competition between the community partition and the scale-free property of the networks. Moreover, we have given the corresponding analysis through the second least eigenvalue λ₂ of the Laplacian matrix of the network which supports the previous result that the synchronization speed is determined by the value of λ₂.     

Explosive synchronization: From synthetic to real-world networks

Synchronization is a widespread phenomenon in both synthetic and real-world networks. This collective behavior of simple and complex systems has been attracting much research during the last decades. Two different routes to synchrony are defined in networks; first-order, characterized as explosive, and second-order, characterized as continuous transition. Although pioneer researches explained that the transition type is a generic feature in the networks, recent studies proposed some frameworks in which different phase and even chaotic oscillators exhibit explosive synchronization. The relationship between the structural properties of the network and the dynamical features of the oscillators is mainly proclaimed because some of these frameworks show abrupt transitions. Despite different theoretical analyses about the appearance of the first-order transition, studies are limited to the mean-field theory, which cannot be generalized to all networks. There are different real-world and man-made networks whose properties can be characterized in terms of explosive synchronization, e.g., the transition from unconsciousness to wakefulness in the brain and spontaneous synchronization of power-grid networks. In this review article, explosive synchronization is discussed from two main aspects. First, pioneer articles are categorized from the dynamical-structural framework point of view. Then, articles that considered different oscillators in the explosive synchronization frameworks are studied. In this article, the main focus is on the explosive synchronization in networks with chaotic and neuronal oscillators. Also, efforts have been made to consider the recent articles which proposed new frameworks of explosive synchronization.