Pinning adaptive synchronization of complex dynamical network with multi-links

Networks with multi-links are universal in the real world such as communication networks, transport networks, and social networks. It is important for us to investigate the control of complex dynamical network with multi-links. In this paper, both local and global stabilities of dynamical network with multi-links are analyzed by applying adaptive control theory and mathematical tools, and some new criteria are proposed to ensure the pinning synchronization. We find that the number of pinned nodes satisfies an inequality for synchronization. Additionally, we solve the problem of how much the coupling strength we need to achieve network synchronization with one pinned node in the network system with multi-links. Finally, numerical examples are used to illustrate the effectiveness of the proposed method.     

Multiple synchronization transitions due to periodic coupling strength in delayed Newman–Watts networks of chaotic bursting neurons

In this paper, we numerically study the effect of time-periodic coupling strength on the synchronization of firing activity in delayed Newman–Watts networks of chaotic bursting neurons. We first examine how the firing synchronization transitions induced by time delay under fixed coupling strength changes in the presence of time-periodic coupling strength, and then focus on how time-periodic coupling strength induces synchronization transitions in the networks. It is found that time delay can induce more synchronization transitions in the presence of time-periodic coupling strength compared to fixed coupling strength. As the frequency of time-periodic coupling strength is varied, the firing exhibits multiple synchronization transitions between spiking antiphase synchronization and in-phase synchronization of various firing behaviors including bursting, spiking, and both bursting and spiking, depending on the values of time delay. These results show that time-periodic coupling strength can increase the synchronization transitions by time delay and can induce multiple synchronization transitions of various firing behaviors in the neuronal networks. This means that time-periodic coupling strength plays an important role in the information processing and transmission in neural systems.     

Adaptive event-triggered prescribed performance learning synchronization for complex dynamical networks with unknown time-varying coupling strength

Nonlinear Dynamics - In this paper, we focus on investigating the prescribed performance synchronization problems of complex dynamical networks with unknown time-varying coupling strength by...     

Cluster synchronization analysis of complex dynamical networks by input-to-state stability

Cluster synchronization is an interesting issue in complex dynamical networks with community structure. In this paper, we study cluster synchronization of complex networks with non-identical systems by input-to-state stability. Some sufficient conditions that ensure cluster synchronization of complex networks are provided. We show that the cluster synchronization is difficult to achieve if there are some links among different clusters. The analysis is then extended to the case where the outer coupling strengths are adaptive. Finally, numerical simulations are given to validate our theoretical analysis.     

Cluster synchronization and rhythm dynamics in a complex neuronal network with chemical synapses

Cluster synchronization and rhythm dynamics are studied for a complex neuronal network with the small world structure connected by chemical synapses. Cluster synchronization is considered as that in-phase burst synchronization occurs inside each group of the network but diversity may take place among different groups. It is found that both one-cluster and multi-cluster synchronization may exist for chemically excitatory coupled neuronal networks, however, only multi-cluster synchronization can be achieved for chemically inhibitory coupled neuronal networks. The rhythm dynamics of bursting neurons can be described by a quantitative characteristic, the width factor. We also study the effects of coupling schemes, the intrinsic property of neurons and the network topology on the rhythm dynamics of the small world neuronal network. It is shown that the short bursting type is robust with respect to the coupling strength and the coupling scheme. As for the network topology, more links can only change the type of long bursting neurons, and short bursting neurons are also robust to the link numbers.     

Complex network analysis of forced synchronization in a hydrodynamically self-excited jet

Previous experiments by Li and Juniper (2013) have shown that a hydrodynamically self-excited jet can synchronize with external acoustic forcing via one of two possible routes: a saddle-node (SN) bifurcation or a torus-death (TD) bifurcation. In this study, we use complex networks to analyze and forecast these two routes to synchronization in a prototypical self-excited flow – an axisymmetric low-density jet at an operating condition close to its first Hopf point. We build the complex networks using two different methods: the visibility algorithm and the recurrence condition. We find that the networks built with the visibility algorithm are high-clustering, hierarchical, and assortative in the degree of their vertices, although only the TD networks are scale free. Nevertheless, we find that the assortativity coefficient is a sufficiently sensitive indicator by which to distinguish between the SN and TD routes to synchronization and to forecast the onset of synchronization. As for the networks built with the recurrence condition, we find that their topological features differ between the two routes to synchronization, but vary predictably along either route. We quantify these variations using statistical measures such as the mean degree, spectral radius, and transitivity dimension. This study shows that complex networks can be a useful tool for distinguishing between the SN and TD routes to synchronization, and for forecasting the proximity of a system to its synchronization boundaries. These findings could open up new opportunities for complex networks to be used in the development of open-loop control strategies for hydrodynamically self-excited flows.     

Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix

In this paper, we investigate the cluster synchronization problem for networks with nonlinearly coupled nonidentical dynamical systems and asymmetrical coupling matrix by using pinning control. We derive sufficient conditions for cluster synchronization for any initial values through a feedback scheme and propose an adaptive feedback algorithm that adjusts the coupling strength. Some numerical examples are then given to illustrate the theoretical results.     

Noise-induced outer synchronization between two different complex dynamical networks

In this paper, based on the theory of stochastic differential equations, we study the outer synchronization between two different complex dynamical networks with noise coupling. The theoretical result shows that two different complex networks can achieve generalized outer synchronization only with white-noise-based coupling. Numerical examples further verify the effectiveness and feasibility of the theoretical results. Numerical evidence shows that the synchronization rate is proportional to the noise intensity.     

Adaptive synchronization of T-S fuzzy complex networks with time-varying delays via the pinning control method

In this paper, the synchronization of Takagi–Sugeno (T-S) fuzzy complex networks with time-varying delays and adaptive coupling weights is studied. Using the pinning control and adaptive feedback strategy, a new general class of complex networks with fuzzy logic is proposed and its synchronization is investigated in terms of linear matrix inequalities (LMIs). The adaptive update law of coupling weight is only related to the dynamical behaviors of directly connected nodes. Based on the Lyapunov stability theory, it is proven that the synchronization of the addressed network can be achieved under those control strategies. Finally, two numerical examples are given to verify the effectiveness of our theoretical results.     

Pinning synchronization of weighted complex networks with variable delays and adaptive coupling weights

In this paper, the synchronization for time-delayed complex networks with adaptive coupling weights is studied. A pinning strategy and a local adaptive scheme to determine coupling weights and feedback gains are proposed. It is noted that our control strategies only rely on some local information other than the global information of the whole network. Finally, the developed techniques are applied to two complex networks which are respectively synchronized to an unstable equilibrium point and a chaotic attractor.     

Firing synchronization of learning neuronal networks with small-world connectivity

The properties of firing synchronization of learning neuronal networks, electrically and chemically coupled ones, with small-world connectivity are studied. First, the variation properties of synaptic weights are examined. Next the effects of the synaptic learning rate on the properties of firing rate and synchronization are investigated. The influences of the coupling strength and the shortcut probability on synchronization are also explored. It is shown that synaptic learning suppresses over-excitement for the networks, helps synchronization for the electrically coupled neuronal network but destroys synchronization for the chemically coupled one. Both introducing shortcuts and increasing the coupling strength are helpful in improving synchronization of the neuronal networks. The spatio-temporal patterns illustrate and confirm the above results.     

Adaptive coupling synchronization in complex network with uncertain boundary

It is difficult that all the boundaries of chaotic system were estimated precisely; this is why the coupling coefficient cannot be determined beforehand in the problem of synchronization of complex networks. Thus, an estimation of coupling coefficient should be given before designing some controllers. In addition, to realize the synchronization, the estimated coupling coefficient has to be large enough. However, it is not true that the larger the coupling coefficient the better the synchronization is. In fact, a coupling coefficient which is larger than what it needs to be means the energy waste. To overcome this difficulty, in this paper we propose an adaptive coupling method. And a new concept about asymptotic stability is presented. Numerical simulations are implemented on different complex networks. The results indicate that the synchronization can be achieved without a large estimated coefficient.     

Chaotic burst synchronization in heterogeneous small-world neuronal network with noise

Chaotic bursting is a fundamental behavior of neurons. In this paper, global and local burst synchronization is studied in a heterogeneous small-world neuronal network of non-identical Hindmarsh-Rose (HR) neurons with noise. It is found that the network can achieve global burst synchronization much more easily than phase synchronization and nearly complete synchronization. Moreover, local burst synchronized clusters have already formed before global burst synchronization happens. We study the effect of the shortcut-adding probability and the heterogeneity coefficient on local and global burst synchronization of the network and find that the introduction of shortcuts facilitates burst synchronization while the heterogeneity has little effect. Moreover, we study the spatiotemporal pattern of the network and find that there is an optimal coupling strength range in which the periodicity of the network is very apparent.     

Synchronized regions of pinned complex networks: spectral analysis

The synchronization problem for a complex dynamical network is investigated in this paper from a spectral analysis approach. It is assumed that only a small portion of the nodes in the network are chosen to be controlled, known as the pinning control scheme. Some new types of synchronized regions for networks with different node dynamics and inner-coupling structures are discovered, especially for the case of the special chaotic node systems with a stable equilibrium point under fully anti-diagonal and partially anti-diagonal couplings. The eigenvalue distributions of the coupling and control matrices for different types of complex networks are obtained. The effects of the network topology, global coupling strength, pinning density, and pinning strength on the network synchronizability are examined through extensive numerical simulations. It is shown that the synchronizability of the pinned network can be effectively improved by increasing the overall coupling strength, pinning density, and pinning strength for some classes of synchronized regions, whereas too large the pinning density and pinning strength will lead to desynchronization for other classes. It is found that small-world networks are not always easier to synchronize than regular rings, and a denser eigenvalue distribution may not always imply better synchronizability.     

Exponential synchronization for second-order nonlinear systems in complex dynamical networks with time-varying inner coupling via distributed event-triggered transmission strategy

This paper focuses on the exponential synchronization problem of complex dynamical networks (CDNs) with time-varying inner coupling via distributed event-triggered transmission strategy. Information update is driven by properly defined event, which depends on the measurement error. Each node is described as a second-order nonlinear dynamic system and only exchanges information with its neighboring nodes over a directed network. Suppose that the network communication topology contains a directed spanning tree. A sufficient condition for achieving exponential synchronization of second-order nonlinear systems in CDNs with time-varying inner coupling is derived. Detailed theoretical analysis on exponential synchronization is performed by the virtues of algebraic graph theory, distributed event-triggered transmission strategy, matrix inequality and the special Lyapunov stability analysis method. Moreover, the Zeno behavior is excluded as well by the strictly positive sampling intervals based on the upper right-hand Dini derivative. It is noted that the amount of communication among network nodes and network congestion have been significantly reduced so as to avoid the waste of network resources. Finally, a simulation example is given to show the effectiveness of the proposed exponential synchronization criteria.     

Chimera states and synchronization behavior in multilayer memristive neural networks

To study the effect of electromagnetic induction on the spatiotemporal dynamic behavior of neural networks, in this paper, we have mainly studied both the synchronization behavior and the evolution of chimera states (CS) in coupled neural networks. To do this, a multilayer memristive neural network was constructed by selecting the Hindmarsh–Rose neurons as the network nodes, and the effect of electromagnetic induction is introduced by using the cubic flux-controlled memristive model as synapse. For simplicity, the following coupling scheme is adopted: only the coupling connections for the neurons between different layers are considered with memristive synapses, while those neurons in each layer are still bidirectional coupled with the classical electrical synapses. It is found that, by adjusting the coupled strength of electrical synapses and the parameters of memristive synapses, the coexistence behavior of coherent and incoherent states, i.e., the CS, could appear in each layer. It is interesting that the CS are also found in inter-layer memristive synapse network. Furthermore, we have discussed the synchronization behavior in this multilayer memristive neural network, one can find when the whole multilayer network is in a synchronization state, not only the spatiotemporal consistency of the CS in each layer neural networks is observed, but also the memductance of all memristive synapses is completely synchronized. Our results suggest that the electromagnetic induction may play an important role in regulating the dynamic behavior of neural networks, and the introduction of memristive synapse into the biological neural network will provide useful clues for revealing the memory behavior of the neural system in human brain.     

Generalized synchronization of spatiotemporal chaos in a weighted complex network

The nodes of the network are composed of the spatiotemporal chaos systems. The relations between the nodes are built through a weighted connection and the nonlinear terms of the chaos systems themselves are taken as coupling functions. The structure of the coupling functions between the connected nodes and the range of the control gain are obtained based on Lyapunov stability theory. It is proven that generalized chaos synchronization of the weight complex network can be realized even if the coupling strength between the nodes is adopted as any weight value. Subsequently, the catalytic reaction diffusion system which has spatiotemporal chaos behavior is taken as example, and simulation results show the effectiveness of the synchronization principle.     

Synchronization in a fractional-order dynamic network with uncertain parameters using an adaptive control strategy

This paper studies synchronization of all nodes in a fractional-order complex dynamic network. An adaptive control strategy for synchronizing a dynamic network is proposed. Based on the Lyapunov stability theory, this paper shows that tracking errors of all nodes in a fractional-order complex network converge to zero. This simple yet practical scheme can be used in many networks such as small-world networks and scale-free networks. Unlike the existing methods which assume the coupling configuration among the nodes of the network with diffusivity, symmetry, balance, or irreducibility, in this case, these assumptions are unnecessary, and the proposed adaptive strategy is more feasible. Two examples are presented to illustrate effectiveness of the proposed method.     

Complex projective synchronization in drive-response networks coupled with complex-variable chaotic systems

In drive-response complex-variable systems, projective synchronization with respect to a real number, real matrix, or even real function means that drive-response systems evolve simultaneously along the same or inverse direction in a complex plane. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, this paper investigates projective synchronization in drive-response networks of coupled complex-variable chaotic systems with respect to complex numbers, called complex projective synchronization (CPS). The adaptive feedback control method is adopted first to achieve CPS in a general drive-response network. For a special class of drive-response networks, the CPS is achieved via pinning control. Furthermore, a universal pinning control scheme is proposed via the adaptive coupling strength method, several simple and useful criteria for CPS are obtained, and all results are illustrated by numerical examples.     

Adaptive synchronization of uncertain dynamical networks with delayed coupling

We propose a simple scheme for the synchronization of an uncertain complex dynamical network with delayed coupling. Based on the Lyapunov stability theory of functional differential equations, certain controllers can be designed for ensuring the states of uncertain dynamical network with coupling delays to globally asymptotically synchronize by combining the adaptive method and linear feedback with the updated feedback strength. Different update gains η _i will lead to different rates toward synchrony, the choice of which depends on the concrete systems and network models. This strategy can be applied to any complex dynamical network (regular, small-world, scale-free or random). Numerical examples with respectively nearest-neighbor coupling and scale-free structure are given to demonstrate the effectiveness of our presented scheme.