Outer synchronization of complex networks with delay via impulse

Synchronization between the driving network and the responding network (outer synchronization) has attracted increasing attention from various fields of science and engineering. In this paper, we address outer synchronization of complex networks with delays. Both the cases of coupling delay and node delay are considered. Employing the impulsive control method which is simple, efficient, low cost, and easy to implement in practical applications, we obtain some sufficient conditions of outer synchronization. It indicates that outer synchronization can be achieved if the maximal impulsive intervals are less than a critical value. Numerical simulations are also given to demonstrate the effectiveness of the proposed impulsive control scheme.     

Finite-time synchronization for complex dynamical networks with time-varying delays

The finite-time synchronization problem of a class of complex dynamical networks with time-varying delays is addressed in this paper. The network topology is assumed to be directed and weakly connected. By introducing a special zero row-sum matrix and combining the Lyapunov?CKrasovskii functional method and the Kronecker product technique, a sufficient condition is presented, which consist of two simple low-dimensional matrix inequalities. Illustrative example is given to show the feasibility of the proposed method.     

Spatiotemporal chaos synchronization between uncertain complex networks with diverse structures

Spatiotemporal chaos synchronization between uncertain complex networks with diverse structures is investigated. The identification law of unknown parameters and the adaptive law of the configuration matrix element in state equations of network nodes are determined based on stability theory, and the conditions of realizing spatiotemporal chaos synchronization between uncertain complex networks with different structures are discussed and obtained. Further, the Fisher–Kolmogorov system with spatiotemporal chaotic behavior is taken as the nodes of drive and response networks to imitate the experiment. It is found that the synchronization performance between two networks is very stable.     

Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling

This paper investigates the adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling, in which the weights of links between two connected nodes are time varying. By the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization are obtained, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects is designed. The numerical examples are presented to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Finite-time synchronization for complex dynamical networks with hybrid coupling and time-varying delay

In this paper, the finite-time synchronization problem has been investigated for a general array model of dynamical networks with time-varying delay and hybrid coupling. The Lyapunov functional method and linear matrix inequality technique are employed to obtain some synchronization criteria less-conservative and delay-dependent. Here, the coupling configuration matrices are not required to be symmetric or irreducible, neither is their off-diagonal entries assumed to be nonnegative. Moreover, the inner linking matrices are arbitrary real matrices. Finally, numerical examples are given to demonstrate the effectiveness of the proposed synchronous criteria.     

Outer synchronization between uncertain complex networks based on backstepping design

In this paper, the outer synchronization between uncertain networks is investigated using the backstepping design. The adaptive laws of uncertain parameters and the structure of control input in the node of response network are determined based on the stability theory. The outer synchronization between networks can be realized only by putting a control input on one node of response network. Further, the effect of the synchronization principle is verified through a simulation experiment.     

Robust synchronization of uncertain fractional-order chaotic systems with time-varying delay

This paper presents a new technique using a recurrent non-singleton type-2 sequential fuzzy neural network (RNT2SFNN) for synchronization of the fractional-order chaotic systems with time-varying delay and uncertain dynamics. The consequent parameters of the proposed RNT2SFNN are learned based on the Lyapunov–Krasovskii stability analysis. The proposed control method is used to synchronize two non-identical and identical fractional-order chaotic systems, with time-varying delay. Also, to demonstrate the performance of the proposed control method, in the other practical applications, the proposed controller is applied to synchronize the master–slave bilateral teleoperation problem with time-varying delay. Simulation results show that the proposed control scenario results in good performance in the presence of external disturbance, unknown functions in the dynamics of the system and also time-varying delay in the control signal and the dynamics of system. Finally, the effectiveness of proposed RNT2SFNN is verified by a nonlinear identification problem and its performance is compared with other well-known neural networks.     

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 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.     

Parameter identification and synchronization of uncertain general complex networks via adaptive-impulsive control

Multifractal theory has been widely used in kinds of field. In this paper, methods were proposed to study in the power-law auto-correlation and cross-correlation of power operating data based on multifractal detrended analysis (MF-DFA) and multifractal detrended cross-correlation analysis (MF-DXA). We find that both the price and the load time series in California power market and PJM power market exhibit long-term correlation. And the cross-correlation behaviors of the two series in each power market and between the two markets are also analyzed by the method MF-DXA after testing the existence of the cross-correlation of the above power operating data. However, there are some differences in the cross-correlation behaviors between the two markets. It shows that the cross-correlation of the price and the load is significant in every time periods in California 1999 power market, but in the year of 2000 in the same region market, the cross-correlation is insignificant in most time periods. Meanwhile, we conclude that cross-correlation is weaker in the California market than in the PJM market by studying the two consecutive years of the California 1999–2000 and PJM 2001–2002 power markets. We also discuss how the time intervals affect the cross-correlation exponents of the power operating data based on time-delay MF-DXA. An interesting finding is that the biggest cross-correlation exponent of the two series appeared in about 12 days time delay for the PJM 2001 power market and strongest cross-correlation in the California 1999 power market is found in lots of cyclical time intervals.     

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.     

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.