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 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.     

超混沌振荡器的混沌耦合同步

提出了两种应用单变量单向混沌耦合同步超混沌振荡器的同步方案。首先给出了混沌耦合同步方案的数学证明，然后通过数值仿真证实了该混沌耦合同步方案的正确性和有效性。最后通过计算条件李雅谱诺夫指数，比较了不同耦合强度下，两种混沌耦合同步和典型的连续耦合同步的同步性能。计算表明在一定的耦合强度下，混沌耦合比典型的连续耦合同步速度快。     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

Generalized projective synchronization of two coupled complex networks of different sizes

We investigate a new generalized projective synchronization between two complex dynamical networks of different sizes. To the best of our knowledge, most of the current studies on projective synchronization have dealt with coupled networks of the same size. By generalized projective synchronization, we mean that the states of the nodes in each network can realize complete synchronization, and the states of a pair of nodes from both networks can achieve projective synchronization. Using the stability theory of the dynamical system, several sufficient conditions for guaranteeing the existence of the generalized projective synchronization under feedback control and adaptive control are obtained. As an example, we use Chua's circuits to demonstrate the effectiveness of our proposed approach.     

Explosive synchronization of multi-layer complex networks based on inter-layer star network connection

Explosive synchronization (ES) is a first-order transition phenomenon that is ubiquitous in various physical and biological systems. In recent years, researchers have focused on explosive synchronization in a single-layer network, but few in multi-layer networks. This paper proposes a frequency-weighted Kuramoto model in multi-layer complex networks with star connection between layers and analyzes the factors affecting the backward critical coupling strength by both theoretical analysis and numerical validation. Our results show that the backward critical coupling strength of each layer network is influenced by the inter-layer interaction strength and the average degree. The number of network layers, the number of nodes, and the network topology can not directly affect the synchronization of the network. Enhancing the inter-layer interaction strength can prevent the emergence of explosive synchronization and increasing the average degree can promote the generation of explosive synchronization.     

时变时滞耦合两个不同复杂网络的自适应广义同步

针对具有时变时滞耦合的两个不同复杂网络, 研究其广义同步问题. 基于Barbalat引理, 设计非线性自适应控制器使得两个网络获得广义同步. 进一步研究了具有未知拓扑结构的两个网络的广义同步问题. 数值仿真表明了本文方法的有效性.     

复杂网络上的部分同步化:奇异态、遥同步与集团同步

近年来,复杂网络上耦合振子的部分同步化引起了人们极大的关注,其潜在或背后的原因是部分同步化斑图在大脑网络中广泛存在,并很可能与大脑的认知或记忆等功能有密切的联系.本文对这些进展进行简单的总结与归纳,并按照学者们研究的不同侧重点,将其分成三方面来进行介绍,即奇异态、遥同步与集团同步化.着重强调这三种情形各自出现的条件、常用的研究模型、检测的方法以及侧重解释的生物现象等方面.并对它们三者之间的相互关系及今后的研究方向做一些简单的探讨.     

Generalized synchronization between different chaotic maps via dead-beat control

This paper presents a new scheme to achieve generalized synchronization(GS) between different discrete-time chaotic(hyperchaotic) systems.The approach is based on a theorem,which assures that GS is achieved when a structural condition on the considered class of response systems is satisfied.The method presents some useful features:it enables exact GS to be achieved in finite time(i.e.,dead-beat synchronization);it is rigorous,systematic,and straightforward in checking GS;it can be applied to a wide class of chaotic maps.Some examples of GS,including the Grassi-Miller map and a recently introduced minimal 2-D quadratic map,are illustrated.     

结点含时滞的具有零和非零时滞耦合的复杂网络混沌同步

时滞耦合的复杂网络同步已经有大量的研究成果,而网络结点含时滞的无时滞耦合的复杂网络同步的研究工作较少.为使网络模型更接近现实和适用更广的范围,建立了网络结点含时滞,而耦合兼零时滞(无时滞)和非零时滞(有时滞)的复杂网络同步模型.在网络结点上分别设置线性控制器和自适应控制器,研究了其混沌同步问题.利用李雅普诺夫稳定性定理,设计相应的正定函数,分别给出了复杂网络同步的充分条件.最后,为证实同步方案的有效性,选择时滞Logistic函数为结点动力系统,在兼无时滞和有时滞的网络上,给出了线性反馈控制同步误差数值演化趋势.     

节点含时滞的不确定复杂网络的自适应同步研究

研究了节点带有时滞,网络结构已知或者完全未知时的不确定动态网络模型的同步问题.基于李雅普诺夫稳定性理论,并按照参数的已知和未知情况分别设计了复杂网络同步控制器和复杂网络同步自适应控制器,给出了网络同步的充分条件,保证了动态网络渐进同步于任意指定的网络中的单独节点的状态.最后,数值结果表明了方法的有效性. 关键词： 自适应同步 不确定复杂网络 Lyapunov稳定理论     

Cluster projective synchronization of complex networks with nonidentical dynamical nodes

We investigate a new cluster projective synchronization(CPS) scheme in time-varying delay coupled complex dynamical networks with nonidentical nodes.Based on the community structure of the networks,the controllers are designed differently for the nodes in one community,which have direct connections to the nodes in the other communities and the nodes without direct connections to the nodes in the other communities.Some sufficient criteria are derived to ensure the nodes in the same group projectively synchronize and there is also projective synchronization between nodes in different groups.Particularly,the weight configuration matrix is not assumed to be symmetric or irreducible.The numerical simulations are performed to verify the effectiveness of the theoretical results.     

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.     

Pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions

This paper studies pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions. To study a state-feedback pinning-controlled network with N nodes, it first converts the controlled network to an extended network of N+1 nodes without controls. It is shown that the controlled synchronizability of the given network is determined by the real part of the smallest nonzero eigenvalue of the coupling matrix of its extended network when the synchronized region is unbounded; but it is determined by the ratio of the real parts of the largest and the smallest nonzero eigenvalues of the coupling matrix when the synchronized region is bounded. Both theoretical analysis and numerical simulation show that the portion of controlled nodes has no critical values when the synchronized region is unbounded, but it has a critical value when the synchronized region is bounded. In the former case, therefore, it is possible to control the network to achieve synchronization by pinning only one node. In the latter case, the network can achieve controlled synchronization only when the portion of controlled nodes is larger than the critical value.     

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.     

结构与参量不确定的网络与网络之间的混沌同步

进行了结构与参量不确定的网络与网络之间的混沌同步研究．通过设计适当的控制输入,不但实现了两个复杂网络之间的混沌同步,而且网络节点状态方程中的未知参量和网络内部节点之间的耦合强度也被同时确定．通过采用具有调制损耗的CO2激光器的状态方程进行仿真实验,发现网络与网络之间的同步性能非常稳定．     

复杂网络时空混沌同步的Backstepping设计

利用Backstepping设计进行了复杂网络时空混沌的同步研究.首先将实现两个混沌系统同步的Backstepping设计推广到由m个时空混沌系统构成任意结构的复杂网络的同步研究中.进一步依据稳定性理论确定了网络同步时配置系数和控制增益满足的关系.整个网络实现同步仅需要在网络中的一个节点施加控制输入即可.进一步通过仿真实验验证了同步机理的有效性.     

一类节点结构互异的复杂网络的混沌同步

提出了一种实现节点结构互异的复杂网络的混沌同步方法.以异结构混沌系统作为节点构造复杂网络,基于Lyapunov稳定性定理确定了复杂网络中连接节点的耦合函数的形式.以Rssler系统、Coullet系统以及Lorenz系统作为网络节点构成的复杂网络为例,仿真模拟发现,整个复杂网络存在稳定的混沌同步现象.此方法不但可以实现任意混沌系统作为节点的网络混沌同步,而且网络节点数对整个复杂网络同步的稳定性也无影响,因而,具有一定的普适性. 关键词： 混沌同步 复杂网络 异结构 Lyapunov稳定性定理     

Pinning sampled-data synchronization for complex networks with probabilistic coupling delay

We deal with the problem of pinning sampled-data synchronization for a complex network with probabilistic time-varying coupling delay. The sampling period considered here is assumed to be less than a given bound. Without using the Kronecker product, a new synchronization error system is constructed by using the property of the random variable and input delay approach. Based on the Lyapunov theory, a delay-dependent pinning sampled-data synchronization criterion is derived in terms of linear matrix inequalities （LMIs） that can be solved effectively by using MATLAB LMI toolbox. Numerical examples are provided to demonstrate the effectiveness of the proposed scheme.     

双频驱动混沌系统的相同步和广义同步

研究了双频混沌信号驱动的混沌振子的广义同步和相同步问题.发现了反偏向的相同步和正偏向的广义同步，即响应振子可以优先与驱动强度弱的混沌信号达到相同步，而广义同步则先在驱动强的信号和响应振子间建立起来.对这些行为产生的动力学机理进行了详细地分析. 关键词： 相同步 广义同步 条件熵 平均频率