Synchronization transition in networked chaotic oscillators: the viewpoint from partial synchronization

Scalable synchronization cluster in networked chaotic oscillators

Cluster synchronization in synthetic networks of coupled chaotic oscillators is investigated. It is found that despite the asymmetric nature of the network structure, a subset of the oscillators can be synchronized as a cluster while the other oscillators remain desynchronized. Interestingly, with the increase in the coupling strength, the cluster is expanding gradually by recruiting the desynchronized oscillators one by one. This new synchronization phenomenon, which is named "scalable synchronization cluster," is explored theoretically by the method of eigenvector-based analysis, and it is revealed that the scalability of the cluster is attributed to the unique feature of the eigenvectors of the network coupling matrix. The transient dynamics of the cluster in response to random perturbations are also studied, and it is shown that in restoring to the synchronization state, oscillators inside the cluster are stabilized in sequence, illustrating again the hierarchy of the oscillators. The findings shed new light on the collective behaviors of networked chaotic oscillators and are helpful for the design of real-world networks where scalable synchronization clusters are concerned.

The transition to synchronization of networked systems

We study the synchronization properties of a generic networked dynamical system, and show that, under a suitable approximation, the transition to synchronization can be predicted with the only help of eigenvalues and eigenvectors of the graph Laplacian matrix. The transition comes out to be made of a well defined sequence of events, each of which corresponds to a specific clustered state. The network's nodes involved in each of the clusters can be identified, and the value of the coupling strength at which the events are taking place can be approximately ascertained. Finally, we present large-scale simulations which show the accuracy of the approximation made, and of our predictions in describing the synchronization transition of both synthetic and real-world large size networks, and we even report that the observed sequence of clusters is preserved in heterogeneous networks made of slightly non-identical systems.

Hierarchy of partially synchronous states in a ring of coupled identical oscillators

In coupled identical oscillators, complete synchronization has been well formulated; however, partial synchronization still calls for a general theory. In this work, we study the partial synchronization in a ring of N locally coupled identical oscillators. We first establish the correspondence between partially synchronous states and conjugacy classes of subgroups of the dihedral group D_{N}. Then we present a systematic method to identify all partially synchronous dynamics on their synchronous manifolds by reducing a ring of oscillators to short chains with various boundary conditions. We find that partially synchronous states are organized into a hierarchical structure and, along a directed path in the structure, upstream partially synchronous states are less synchronous than downstream ones.

Cluster synchronization induced by manifold deformation

Pinning control of cluster synchronization in a globally connected network of chaotic oscillators is studied. It is found in simulations that when the pinning strength exceeds a critical value, the oscillators are synchronized into two different clusters, one formed by the pinned oscillators and the other one formed by the unpinned oscillators. The numerical results are analyzed by the generalized method of master stability function (MSF), in which it is shown that whereas the method is able to predict the synchronization behaviors of the pinned oscillators, it fails to predict the synchronization behaviors of the unpinned oscillators. By checking the trajectories of the oscillators in the phase space, it is found that the failure is attributed to the deformed synchronization manifold of the unpinned oscillators, which is clearly deviated from that of isolated oscillator under strong pinnings. A similar phenomenon is also observed in the pinning control of cluster synchronization in a complex network of symmetric structures and in the self-organized cluster synchronization of networked neural oscillators. The findings are important complements to the generalized MSF method and provide an alternative approach to the manipulation of synchronization behaviors in complex network systems.

Anticipating measure synchronization in coupled Hamiltonian systems with machine learning

A model-free approach is proposed for anticipating the occurrence of measure synchronization in coupled Hamiltonian systems. Specifically, by the technique of parameter-aware reservoir computing in machine learning, we demonstrate that the machine trained by the time series of coupled Hamiltonian systems at a handful of coupling parameters is able to predict accurately not only the critical coupling for the occurrence of measure synchronization, but also the variation of the system order parameters around the transition point. The capability of the model-free technique in anticipating measure synchronization is exemplified in Hamiltonian systems of two coupled oscillators and also in a Hamiltonian system of three globally coupled oscillators where partial synchronization arises. The studies pave a way to the model-free, data-driven analysis of measure synchronization in large-size Hamiltonian systems.

Matryoshka and disjoint cluster synchronization of networks

The main motivation for this paper is to characterize network synchronizability for the case of cluster synchronization (CS), in an analogous fashion to Barahona and Pecora [Phys. Rev. Lett. 89, 054101 (2002)] for the case of complete synchronization. We find this problem to be substantially more complex than the original one. We distinguish between the two cases of networks with intertwined clusters and no intertwined clusters and between the two cases that the master stability function is negative either in a bounded range or in an unbounded range of its argument. Our proposed definition of cluster synchronizability is based on the synchronizability of each individual cluster within a network. We then attempt to generalize this definition to the entire network. For CS, the synchronous solution for each cluster may be stable, independent of the stability of the other clusters, which results in possibly different ranges in which each cluster synchronizes (isolated CS). For each pair of clusters, we distinguish between three different cases: Matryoshka cluster synchronization (when the range of the stability of the synchronous solution for one cluster is included in that of the other cluster), partially disjoint cluster synchronization (when the ranges of stability of the synchronous solutions partially overlap), and complete disjoint cluster synchronization (when the ranges of stability of the synchronous solutions do not overlap).

Cluster synchronization of networks via a canonical transformation for simultaneous block diagonalization of matrices

We study cluster synchronization of networks and propose a canonical transformation for simultaneous block diagonalization of matrices that we use to analyze the stability of the cluster synchronous solution. Our approach has several advantages as it allows us to: (1) decouple the stability problem into subproblems of minimal dimensionality while preserving physically meaningful information, (2) study stability of both orbital and equitable partitions of the network nodes, and (3) obtain a parameterization of the problem in a small number of parameters. For the last point, we show how the canonical transformation decouples the problem into blocks that preserve key physical properties of the original system. We also apply our proposed algorithm to analyze several real networks of interest, and we find that it runs faster than alternative algorithms from the literature.

Synchronization within synchronization: transients and intermittency in ecological networks

Transients are fundamental to ecological systems with significant implications to management, conservation and biological control. We uncover a type of transient synchronization behavior in spatial ecological networks whose local dynamics are of the chaotic, predator–prey type. In the parameter regime where there is phase synchronization among all the patches, complete synchronization (i.e. synchronization in both phase and amplitude) can arise in certain pairs of patches as determined by the network symmetry—henceforth the phenomenon of  ‘synchronization within synchronization.’ Distinct patterns of complete synchronization coexist but, due to intrinsic instability or noise, each pattern is a transient and there is random, intermittent switching among the patterns in the course of time evolution. The probability distribution of the transient time is found to follow an algebraic scaling law with a divergent average transient lifetime. Based on symmetry considerations, we develop a stability analysis to understand these phenomena. The general principle of symmetry can also be exploited to explain previously discovered, counterintuitive synchronization behaviors in ecological networks.

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.

Stable Chimeras and Independently Synchronizable Clusters

Cluster synchronization is a phenomenon in which a network self-organizes into a pattern of synchronized sets. It has been shown that diverse patterns of stable cluster synchronization can be captured by symmetries of the network. Here, we establish a theoretical basis to divide an arbitrary pattern of symmetry clusters into independently synchronizable cluster sets, in which the synchronization stability of the individual clusters in each set is decoupled from that in all the other sets. Using this framework, we suggest a new approach to find permanently stable chimera states by capturing two or more symmetry clusters-at least one stable and one unstable-that compose the entire fully symmetric network.

Inducing isolated-desynchronization states in complex network of coupled chaotic oscillators

In a recent study about chaos synchronization in complex networks [Nat. Commun. 5, 4079 (2014)NCAOBW2041-172310.1038/ncomms5079], it is shown that a stable synchronous cluster may coexist with vast asynchronous nodes, resembling the phenomenon of a chimera state observed in a regular network of coupled periodic oscillators. Although of practical significance, this new type of state, namely, the isolated-desynchronization state, is hardly observed in practice due to its strict requirements on the network topology. Here, by the strategy of pinning coupling, we propose an effective method for inducing isolated-desynchronization states in symmetric networks of coupled chaotic oscillators. Theoretical analysis based on eigenvalue analysis shows that, by pinning a group of symmetric nodes in the network, there exists a critical pinning strength beyond which the group of pinned nodes can completely be synchronized while the unpinned nodes remain asynchronous. The feasibility and efficiency of the control method are verified by numerical simulations of both artificial and real-world complex networks with the numerical results in good agreement with the theoretical predictions.

Reviving oscillation with optimal spatial period of frequency distribution in coupled oscillators

The spatial distributions of system's frequencies have significant influences on the critical coupling strengths for amplitude death (AD) in coupled oscillators. We find that the left and right critical coupling strengths for AD have quite different relations to the increasing spatial period m of the frequency distribution in coupled oscillators. The left one has a negative linear relationship with m in log-log axis for small initial frequency mismatches while remains constant for large initial frequency mismatches. The right one is in quadratic function relation with spatial period m of the frequency distribution in log-log axis. There is an optimal spatial period m₀ of frequency distribution with which the coupled system has a minimal critical strength to transit from an AD regime to reviving oscillation. Moreover, the optimal spatial period m₀ of the frequency distribution is found to be related to the system size N. Numerical examples are explored to reveal the inner regimes of effects of the spatial frequency distribution on AD.

Controlling synchronous patterns in complex networks

Although the set of permutation symmetries of a complex network could be very large, few of them give rise to stable synchronous patterns. Here we present a general framework and develop techniques for controlling synchronization patterns in complex network of coupled chaotic oscillators. Specifically, according to the network permutation symmetry, we design a small-size and weighted network, namely the control network, and use it to control the large-size complex network by means of pinning coupling. We argue mathematically that for any of the network symmetries, there always exists a critical pinning strength beyond which the unstable synchronous pattern associated to this symmetry can be stabilized. The feasibility of the control method is verified by numerical simulations of both artificial and real-world networks and demonstrated experimentally in systems of coupled chaotic circuits. Our studies show the controllability of synchronous patterns in complex networks of coupled chaotic oscillators.

Experimental Study of the Triplet Synchronization of Coupled Nonidentical Mechanical Metronomes

Triplet synchrony is an interesting state when the phases and the frequencies of three coupled oscillators fulfill the conditions of a triplet locking, whereas every pair of systems remains asynchronous. Experimental observation of triplet synchrony is firstly realized in three coupled nonidentical mechanical metronomes. A more direct method based on the phase diagram is proposed to observe and determine triplet synchronization. Our results show that the stable triplet synchrony is observed in several intervals of the parameter space. Moreover, the experimental results are verified according to the theoretical model of the coupled metronomes. The outcomes are useful to understand the inner regimes of collective dynamics in coupled oscillators.

Cyclic synchronous patterns in coupled discontinuous maps

Cyclic collective behaviors are commonly observed in biological and neuronal systems, yet the dynamical origins remain unclear. Here, by models of coupled discontinuous map lattices, we investigate the cyclic collective behaviors by means of cluster synchronization. Specifically, we study the synchronization behaviors in lattices of coupled periodic piecewise-linear maps and find that in the nonsynchronous regime the maps can be synchronized into different clusters and, as the system evolves, the synchronous clusters compete with each other and present the recurring process of cluster expanding, shrinking, and switching, i.e., showing the cyclic synchronous patterns. The dynamical mechanisms of cyclic synchronous patterns are explored, and the crucial roles of basin distribution are revealed. Moreover, due to the discontinuity feature of the map, the cyclic patterns are found to be very sensitive to the system initial conditions and parameters, based on which we further propose an efficient method for controlling the cyclic synchronous patterns.

Consistency between functional and structural networks of coupled nonlinear oscillators

In data-based reconstruction of complex networks, dynamical information can be measured and exploited to generate a functional network, but is it a true representation of the actual (structural) network? That is, when do the functional and structural networks match and is a perfect matching possible? To address these questions, we use coupled nonlinear oscillator networks and investigate the transition in the synchronization dynamics to identify the conditions under which the functional and structural networks are best matched. We find that, as the coupling strength is increased in the weak-coupling regime, the consistency between the two networks first increases and then decreases, reaching maximum in an optimal coupling regime. Moreover, by changing the network structure, we find that both the optimal regime and the maximum consistency will be affected. In particular, the consistency for heterogeneous networks is generally weaker than that for homogeneous networks. Based on the stability of the functional network, we propose further an efficient method to identify the optimal coupling regime in realistic situations where the detailed information about the network structure, such as the network size and the number of edges, is not available. Two real-world examples are given: corticocortical network of cat brain and the Nepal power grid. Our results provide new insights not only into the fundamental interplay between network structure and dynamics but also into the development of methodologies to reconstruct complex networks from data.

Control for a synchronization-desynchronization switch

How to freely enhance or suppress synchronization of networked dynamical systems is of great importance in many disciplines. A unified precise control method for a synchronization-desynchronization switch, called the pull-push control method, is suggested. Namely, synchronization can be achieved when the original systems are desynchronous by pulling (or protecting) one node or a certain subset of nodes, whereas desynchronization can be accomplished when the systems are already synchronous by pushing (or kicking) one node or a certain subset of nodes. With this method, the controlled nodes should be chosen by the generalized eigenvector centrality of the critical synchronization mode of the Laplacian matrix. Compared with existing control methods for synchronization, it displays high efficiency, flexibility, and precision as well.