Narrow Results

The stability and controllability of asymmetric complex dynamical networks are investigated in detail based on eigenvalue analysis. Pinning control is suggested to stabilize the homogenous stationary state of the whole coupled network. The complicated coupled problem is reduced to two independent problems: clarifying the stable regions of the coupled network and specifying the eigenvalue distribution of the asymmetric coupling and control matrices. The dependence of the controllability on both pinning density and pinning strength is studied.

This paper concerns the outer synchronization problem between two complex delayed networks via the method of aperiodically intermittent pinning control. Apart from previous works, internal delay and coupling delay are both involved in this model, and the designed intermittent controllers can be aperiodic. The main work in this paper can be summarized as follows: First, two cases of aperiodically intermittent control with constant gain and adaptive gain are implemented, respectively. The intermittent control and pinning control are combined to reduce consumptions further. Then, based on the Lyapunov stability theory, synchronization protocols are given by strict derivation. Especially, the designed controllers are indeed simple and valid in application of theory to practice. Finally, numerical examples put the proposed control methods to the test.

In most conventional complex network equilibrium-pinning control, all the network nodes are usually balanced by attaching at least one controller to each node of the concerned network. When the node number is huge and the inter-node connections are complex, this control strategy requires the ability to manipulate the inter-node coupling strength to grow rapidly and intensively. In practical applications, however, the inter-node coupling strength cannot be increased unlimitedly; as a matter of fact, the coupling strength cannot be further changed after reaching some saturation threshold. In this paper, by exploiting the improved coupling strength saturation function, we suggest a new pinning control strategy that makes the network reach its equilibrium-pinning point more effectively than the network with saturated coupling strength. Numerical examples are illustrated to show the effectiveness of the main results.

In this paper, group consensus of second-order multi-agent systems with nonlinear dynamics is investigated. First, we design the distributed protocols for achieving group consensus, in which the strengths of the interactions among the agents are enhanced through tuning the coupling strengths. Further, taking the difference of the edges among agents into account, edge-based distributed protocols through tuning coupling weights of a fraction of edges are designed. Remarkably, only the edges of spanning tree in each group are pinned and the coupling strengths or weights of pinned edges are enhanced according to the updated laws. Both the types of distributed protocols are proved analytically and verified by numerical illustrations.

Recently, synchronization of complex networks has been studied deeply; however, most existing works take the standard proportional control strategy approach to pinning controllers, which produces a steady-state error and has a significant time of fluctuation. In order to solve this problem, we propose a new method using the Proportional-Integral-Differential (PID) control to design a pinning control method for synchronization of complex networks. Based on Lyapunov theory, the stability of the network is proved, and sufficient synchronization conditions are obtained. Finally, examples are given to evaluate the effectiveness of the proposed PID pinning method and compare it to the traditional proportional pinning controller and adaptive pinning controller.

The evolutionary game on graphs provides a natural framework to investigate the cooperation behavior existing in natural and social society. In this paper, degree-based pinning control and random pinning control are introduced into the evolutionary prisoner's dilemma game on scale-free networks, and the effects of control mechanism and control cost on the evolution are studied. Numerical simulation shows that forcing some nodes to cooperate (defect) will increase (decrease) the frequency of cooperators. Compared with random pinning control, degree-based pinning control is more efficient, and degree-based pinning control costs less than random pinning control to achieve the same goal. Numerical results also reveal that the evolutionary time series is more stable under pinning control mechanisms, especially under the degree-based pinning control.

In this paper, a modified degree-based strategy is proposed for pinning control of complex networks under constrained control capacity. Limited by the total control strength, the pinning strategy design problem is reconsidered as a constrained optimization problem. To provide a fast and effective solution, a decrease-and-conquer approach has been designed to assign the best control strengths to the pinned highest-degree nodes. Simulation results have demonstrated that the obtained strategy provides an outperformed control effect on various kinds of complex networks, as compared with strategies derived from degree, betweenness or closeness using evenly-distributed control strength.

The local stochastic stability of complex networks under pinning control is studied, with stochastic perturbations to the coupling strengths. The nodes of complex network are modeled as second-order differential equations subject to stochastic parametric excitations. The complex network is first linearized at its trivial solution, and the resulting equations are reduced to independent subsystems by using a suitable linear transformation. Then the condition of the stochastic stability can be determined by Lyapunov exponents of the subsystems. The largest Lyapunov exponent of the subsystem can be expressed analytically, as a function of the eigenvalue of a matrix associated with the coupling matrix and pinning control matrix for a given system of parameters. And the stability region with respect to the eigenvalue can be obtained. It is pointed out that the local stochastic stability of the network is finally determined by the maximal and minimum eigenvalues of the matrix. Numerical results with positive and negative damping coefficients are given to illustrate the criterion. Moreover, for the positive damping coefficient case, the pinning control may destabilize the network; and for the negative damping coefficient case, the pinning control may stabilize the network.

In this paper, the problem of pinning impulsive synchronization for complex dynamical networks with directed or undirected but a strongly connected topology is investigated. To remedy this problem, we propose an efficient algorithm to find certain suitable nodes to be controlled via pinning and these selected nodes, in general, could be different at distinct impulsive time instants. The proposed algorithm guarantees the efficiency of the designed pinning impulsive strategy for the global exponential synchronization of state-coupled dynamical networks under an easily-verified condition. In other words, impulsive controllers and an efficient algorithm are designed to control a small fraction of the nodes, which successfully controls the whole dynamical network. Furthermore, we also estimate the upper bound of the number of pinning nodes, which is shown to be closely related to the impulsive intervals. The relationship implies that the required number of pinning nodes, which should be controlled for the successful control of the whole dynamical network, can be greatly reduced by reducing the impulses interval. Finally, simulations of scale-free and small-world networks are given to illustrate the effectiveness of the theoretical results.

In this work, we present two different hybrid pinning strategies to synchronize a complex network of identical agents into a known desired solution. The first strategy is the chaos control hybrid pinning in which pinning synchronization control and chaos control are merged. The second strategy is the nonidentical reference hybrid pinning, in which the pinning reference dynamical behavior is different from network nodes dynamical behavior.

In this paper, we aim to identify a directed complex network with optimal controllability, for which pinning control is to be applied. Since the controllability of a network can be reflected by the smallest nonzero eigenvalue of a matrix related to its topology, an optimal network design problem is formulated based on the maximization of this eigenvalue. To better consider the practical reality, constraints on node degree sequence are specified. Based on the derived bounds of the eigenvalue, an effective rewiring scheme is designed and solutions close to or equal to the upper bound are obtained. Finally, the relationship between network characteristics and controllability is studied. Through complexity analysis, it is concluded that the network with high controllability should possess two properties, i.e. nodes with high out-degree for pinning and other nodes with uniform degree distribution.

In this paper, a control strategy based on the inverse optimal control approach is applied for pinning weighted complex networks with chaotic systems at their nodes; additionally, a cost functional is minimized. This control strategy does not require to have the same coupling strength for all node connections.

Large scale networks may exhibit complicated dynamical behaviors, such as instability, bifurcation, and chaos. It is not easy to get which nodes or links need to be controlled for ensuring the desired dynamical behavior. This paper presents a detailed analysis on the dynamics and control of a delayed network by a pinning strategy. The network system is first mapped to a simple system by means of an orthogonal transformation. The control signals are then exerted only on a fraction of nodes in the transformed system. We analyze how to get the desired dynamics of the original controlled system (such as zeros solutions, periodic solutions, quasi-periodic solutions and chaos) via controlling the transformed system. It shows that the solutions of the original system can be obtained by the local dynamical behavior exhibiting in the transformed system. The results show that the proposed pinning control strategy is an effective approach for dynamics control.

We study synchronization through the combination of electrical synapse and hybrid synapse, and analyse frequency responses in a network constituted of two subnetworks (layers). We prove the onset of synchronization for two coupled nondelayed neurons when we set properly the coupling strength of electrical and chemical synapses, and neurotransmitter parameters of chemical synapse with constant initial conditions. The chemical synapse takes into consideration inhibitory and excitatory behaviors of the neuron. We further prove that for random initial conditions of few sets of identical nodes which constitute a network with two layers, global synchronization is possible for specific values of coupling strength and inner matrix configuration. To ensure great accuracy of the global synchronization, we add a controller and for all settings of electrical and chemical parameters, we always observe global synchronization in both layers and then, the associated network. Five different network topologies were used for the analysis. We finally verify the robustness of the global synchronization by using an external sinusoidal stimuli and observe that the control action is very powerful to maintain the global synchronization dynamic of the network. We observe equally that the instantaneous frequency can toggle between delta oscillations, awake state of the brain, up to gamma oscillations which are the frequencies of intensive activities of the brain.

The synchronization problem of a kind of complex dynamical networks with uncertain links and external disturbances is studied in this paper. The adaptive and impulsive controller is designed after a series of researches on nonlinearity of joints, time variability and uncertainty of system parameters as well as uncertainty of coupling relationship among joints. The stability criteria of the uncertain complex dynamical networks are obtained based on the robust control theory and Lyapunov function theory, thus the synchronization for this kind of complex networks can be realized. The simulation results verify the effectiveness of the method in this paper.