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.     

Explosive synchronization of complex networks with different chaotic oscillators

Recent studies have shown that explosive synchronization transitions can be observed in networks of phase oscillators [Gómez-Garden es J,Gómez S,Arenas A and Moreno Y 2011 Phys.Rev.Lett.106 128701] and chaotic oscillators [Leyva I,Sevilla-Escoboza R,BuldúJ M,Sendin a-Nadal I,Gómez-Garden es J,Arenas A,Moreno Y,Gómez S,Jaimes-Reátegui R and Boccaletti S 2012 Phys.Rev.Lett.108 168702].Here,we study the effect of different chaotic dynamics on the synchronization transitions in small world networks and scale free networks.The continuous transition is discovered for Rssler systems in both of the above complex networks.However,explosive transitions take place for the coupled Lorenz systems,and the main reason is the abrupt change of dynamics before achieving complete synchronization.Our results show that the explosive synchronization transitions are accompanied by the change of system dynamics.     

具有随机节点结构的复杂网络同步研究

针对具有随机节点结构的复杂网络, 研究其同步问题. 基于Lyapunov稳定性理论和线性矩阵不等式技术给出了复杂网络同步稳定的充分性条件, 该充分性条件不仅与复杂网络的状态时延有关, 还与节点结构的概率分布有关. 数值仿真表明本文方法的有效性. 关键词： 复杂网络 随机节点 同步稳定 时滞     

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.     

Cluster synchronization in fractional-order complex dynamical networks

Cluster synchronization of complex dynamical networks with fractional-order dynamical nodes is discussed in the Letter. By using the stability theory of fractional-order differential system and linear pinning control, a sufficient condition for the stability of the synchronization behavior in complex networks with fractional order dynamics is derived. Only the nodes in one community which have direct connections to the nodes in other communities are needed to be controlled, resulting in reduced control cost. A numerical example is presented to demonstrate the validity and feasibility of the obtained result. Numerical simulations illustrate that cluster synchronization performance for fractional-order complex dynamical networks is influenced by inner-coupling matrix, control gain, coupling strength and topological structures of the networks.     

Robust adaptive synchronization of uncertain and delayed dynamical complex networks with faulty network

<正>This paper presents a new robust adaptive synchronization method for a class of uncertain dynamical complex networks with network failures and coupling time-varying delays.Adaptive schemes are proposed to adjust controller parameters for the faulty network compensations,as well as to estimate the upper and lower bounds of delayed state errors and perturbations to compensate the effects of delay and perturbation on-line without assuming symmetry or irreducibility of networks.It is shown that,through Lyapunov stability theory,distributed adaptive controllers constructed by the adaptive schemes are successful in ensuring the achievement of asymptotic synchronization of networks in the present of faulty and delayed networks,and perturbation inputs.A Chua's circuit network example is finally given to show the effectiveness of the proposed synchronization criteria.     

Stochastic synchronization for time-varying complex dynamical networks

This paper studies the stochastic synchronization problem for time-varying complex dynamical networks. This model is totally different from some existing network models. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, some controllers and adaptive laws are designed to ensure achieving stochastic synchronization of a complex dynamical network model. A sufficient synchronization condition is given to ensure that the proposed network model is mean-square stable. Theoretical analysis and numerical simulation fully verify the main results.     

Cluster synchronization of uncertain complex networks with desynchronizing impulse

Cluster synchronization of nonlinear uncertain complex networks with desynchronizing impulse is explored. First of all, a feedback controller is employed, based on the Lyapunov stability theorem and Lipschitz condition, to guarantee that the uncertain complex networks with desynchronizing impulse synchronize with an object trajectory. Furthermore, a synchronizing impulse controller is presented, which is more efficiently and directly used to achieve the cluster synchronization. Finally, numerical examples are examined to show the effectiveness of the proposed methods.     

Explosive synchronization in a mobile network in the presence of a positive feedback mechanism

Synchronization is a process that describes the coherent dynamics of a large ensemble of interacting units.The study of explosive synchronization transition attracts considerable attention.Here,I report the explosive transition within the framework of a mobile network,while each oscillator is controlled by global-order parameters of the system.Using numerical simulation,I find that the explosive synchronization(ES)transition behavior can be controlled by simply adjusting the fraction of controlled oscillators.The influences of some parameters on explosive synchronization are studied.Moreover,due to the presence of the positive feedback mechanism,I prevent the occurrence of the synchronization of continuous-phase transition and make phase transition of the system a first-order phase transition accompanied by a hysteresis loop.     

Adaptive pinning synchronization in fractional-order complex dynamical networks

Synchronization of general complex dynamical networks with fractional-order dynamical nodes is addressed in this paper. Based on the stability theory of fractional-order differential systems and adaptive pinning control, some sufficient local asymptotical synchronization criteria and global asymptotical ones are derived respectively, which succeed in solving the problem about how many nodes are need to be controlled and how much coupling strength should be applied to ensure the synchronization of the entire fractional-order networks. The obtained results are more general and effective than those reported. Moreover, the coupling-configuration matrices and the inner-coupling matrices are not assumed to be symmetric and irreducible. Finally, a numerical simulation is presented to demonstrate the validity and feasibility of the proposed synchronization criteria.     

融合复杂动态网络的模型参考自适应同步研究

本文根据融合复杂网络边性质的不同, 运用网络拆分的思想研究了多重边融合复杂网络的自适应同步问题.基于Lyapunov稳定性理论,采用自适应反馈控制方法,在网络节点相同和不同的情况下,分别给出了网络全局同步的准则以及相应的控制器.最后,数值仿真验证了本文方法的有效性. 关键词： 融合网络 自适应同步 Lyapunov稳定性理论     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.     

基于低阶矩阵最大特征值的复杂网络牵制混沌同步

虽已对复杂网络牵制同步需要牵制结点数量及牵制结点数量与耦合强度的关系进行了研究,然而快速计算牵制结点数量仍是大规模复杂网络面临的一个重要问题.研究发现了复杂网络耦合矩阵主子阵最大值递减规律,由此提出了快速计算复杂网络牵制结点数量的方法,揭示了不同的牵制策略与牵制结点数量之间的关系.数值仿真显示了在无标度网络和小世界网络上三种不同的牵制策略下,牵制结点数与主子阵最大特征值的变化规律;最后给出了一个在无标度网络上采用随机选择结点策略的牵制同步实例.     

Adaptive function projective synchronization of uncertain complex dynamical networks with disturbance

We investigate the problem of function projective synchronization (FPS) in drive-response dynamical networks with non-identical nodes. An adaptive controller is proposed for the FPS of complex dynamical networks with uncertain parameters and disturbance. Not only are the unknown parameters of the networks estimated by the adaptive laws obtained from the Lyapunov stability theory and Taylor expansions, but the unknown bounded disturbances are also simultaneously conquered by the proposed control. Finally, a numerical simulation is provided to illustrate the feasibility and effectiveness of the obtained result.     

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.     

The synchronization transition in correlated oscillator populations

The synchronization transition of correlated ensembles of coupled Kuramoto oscillators on sparse random networks is investigated. Extensive numerical simulations show that correlations between the native frequencies of adjacent oscillators on the network systematically shift the critical point as well as the critical exponents characterizing the transition. Negative correlations imply an onset of synchronization for smaller coupling, whereas positive correlations shift the critical coupling towards larger interaction strengths. For negatively correlated oscillators the transition still exhibits critical behaviour similar to that of the all-to-all coupled Kuramoto system, while positive correlations change the universality class of the transition depending on the correlation strength. Crucially, the paper demonstrates that the synchronization behaviour is not only determined by the coupling architecture, but also strongly influenced by the oscillator placement on the coupling network.     

Outer synchronization between two different fractional-order general complex dynamical networks

Outer synchronization between two different fractional-order general complex dynamical networks is investigated in this paper.Based on the stability theory of the fractional-order system,the sufficient criteria for outer synchronization are derived analytically by applying the nonlinear control and the bidirectional coupling methods.The proposed synchronization method is applicable to almost all kinds of coupled fractional-order general complex dynamical networks.Neither a symmetric nor irreducible coupling configuration matrix is required.In addition,no constraint is imposed on the inner-coupling matrix.Numerical examples are also provided to demonstrate the validity of the presented synchronization scheme.Numeric evidence shows that both the feedback strength k and the fractional order α can be chosen appropriately to adjust the synchronization effect effectively.     

New synchronization analysis for complex networks with variable delays

This paper deals with the issue of synchronization of delayed complex networks. Differing from previous results,the delay interval [0,d(t)] is divided into some variable subintervals by employing a new method of weighting delays. Thus,new synchronization criteria for complex networks with time-varying delays are derived by applying this weighting-delay method and introducing some free weighting matrices. The obtained results have proved to be less conservative than previous results. The sufficient conditions of asymptotical synchronization are derived in the form of linear matrix inequality,which are easy to verify. Finally,several simulation examples are provided to show the effectiveness of the proposed results.     

Ordered bursting synchronization and complex wave propagation in a ring neuronal network

Ordered bursting synchronization and complex propagation are investigated for a ring neuronal network in which each neuron exhibits chaotic bursting behaviour. The neurons become more and more synchronous in chaotic bursting as the synaptic strength is increased. It is shown that excitatory chemical synapses can effectively tame the chaos, and ordered bursting synchronization can be observed as the synaptic strength is further increased. However, synchronization among neurons is weakened as the number of neurons is increased. More importantly, it is shown that ordered bursting synchronization can be turned into spiking synchronization at certain noise intensity. Complex spatio-temporal patterns propagating towards both sides of pacemaker are found in this network before the emergence of spiking synchronization.