Distributed synchronization of coupled neural networks via randomly occurring control

In this paper, we study the distributed synchronization and pinning distributed synchronization of stochastic coupled neural networks via randomly occurring control. Two Bernoulli stochastic variables are used to describe the occurrences of distributed adaptive control and updating law according to certain probabilities. Both distributed adaptive control and updating law for each vertex in a network depend on state information on each vertex's neighborhood. By constructing appropriate Lyapunov functions and employing stochastic analysis techniques, we prove that the distributed synchronization and the distributed pinning synchronization of stochastic complex networks can be achieved in mean square. Additionally, randomly occurring distributed control is compared with periodically intermittent control. It is revealed that, although randomly occurring control is an intermediate method among the three types of control in terms of control costs and convergence rates, it has fewer restrictions to implement and can be more easily applied in practice than periodically intermittent control.

Finite-time mixed outer synchronization of complex networks with coupling time-varying delay

This article is concerned with the problem of finite-time mixed outer synchronization (FMOS) of complex networks with coupling time-varying delay. FMOS is a recently developed generalized synchronization concept, i.e., in which different state variables of the corresponding nodes can evolve into finite-time complete synchronization, finite-time anti-synchronization, and even amplitude finite-time death simultaneously for an appropriate choice of the controller gain matrix. Some novel stability criteria for the synchronization between drive and response complex networks with coupling time-varying delay are derived using the Lyapunov stability theory and linear matrix inequalities. And a simple linear state feedback synchronization controller is designed as a result. Numerical simulations for two coupled networks of modified Chua's circuits are then provided to demonstrate the effectiveness and feasibility of the proposed complex networks control and synchronization schemes and then compared with the proposed results and the previous schemes for accuracy.

Synchronization error estimation and controller design for delayed Lur'e systems with parameter mismatches

This paper investigates the problem of master-slave synchronization of two delayed Lur'e systems in the presence of parameter mismatches. First, by analyzing the corresponding synchronization error system, synchronization with an error level, which is referred to as quasi-synchronization, is established. Some delay-dependent quasi-synchronization criteria are derived. An estimation of the synchronization error bound is given, and an explicit expression of error levels is obtained. Second, sufficient conditions on the existence of feedback controllers under a predetermined error level are provided. The controller gains are obtained by solving a set of linear matrix inequalities. Finally, a delayed Chua's circuit is chosen to illustrate the effectiveness of the derived results.

Filippov systems and quasi-synchronization control for switched networks

This paper is concerned with the quasi-synchronization issue of linearly coupled networks with discontinuous nonlinear functions in each isolated node. Under the framework of Filippov systems, the existence and boundedness of solutions for such complex networks can be guaranteed by the matrix measure approach. A design method is presented for the synchronization controllers of coupled networks with non-identical discontinuous systems. Moreover, a sufficient condition is derived to ensure the quasi-synchronization of switched coupled complex networks with discontinuous isolated nodes, which could be controlled by some designed linear controllers. The obtained results extend the previous work on the synchronization issue of coupled complex networks with Lipschitz continuous conditions. Numerical simulations on the coupled chaotic systems are given to demonstrate the effectiveness of the theoretical results.

Transition to complete synchronization and global intermittent synchronization in an array of time-delay systems

We report the nature of transitions from the nonsynchronous to a complete synchronization (CS) state in arrays of time-delay systems, where the systems are coupled with instantaneous diffusive coupling. We demonstrate that the transition to CS occurs distinctly for different coupling configurations. In particular, for unidirectional coupling, locally (microscopically) synchronization transition occurs in a very narrow range of coupling strength but for a global one (macroscopically) it occurs sequentially in a broad range of coupling strength preceded by an intermittent synchronization. On the other hand, in the case of mutual coupling, a very large value of coupling strength is required for local synchronization and, consequently, all the local subsystems synchronize immediately for the same value of the coupling strength and, hence, globally, synchronization also occurs in a narrow range of the coupling strength. In the transition regime, we observe a type of synchronization transition where long intervals of high-quality synchronization which are interrupted at irregular times by intermittent chaotic bursts simultaneously in all the systems and which we designate as global intermittent synchronization. We also relate our synchronization transition results to the above specific types using unstable periodic orbit theory. The above studies are carried out in a well-known piecewise linear time-delay system.

Robust synchronization for 2-D discrete-time coupled dynamical networks

In this paper, a new synchronization problem is addressed for an array of 2-D coupled dynamical networks. The class of systems under investigation is described by the 2-D nonlinear state space model which is oriented from the well-known Fornasini-Marchesini second model. For such a new 2-D complex network model, both the network dynamics and the couplings evolve in two independent directions. A new synchronization concept is put forward to account for the phenomenon that the propagations of all 2-D dynamical networks are synchronized in two directions with influence from the coupling strength. The purpose of the problem addressed is to first derive sufficient conditions ensuring the global synchronization and then extend the obtained results to more general cases where the system matrices contain either the norm-bounded or the polytopic parameter uncertainties. An energy-like quadratic function is developed, together with the intensive use of the Kronecker product, to establish the easy-to-verify conditions under which the addressed 2-D complex network model achieves global synchronization. Finally, a numerical example is given to illustrate the theoretical results and the effectiveness of the proposed synchronization scheme.

Complete synchronization of Boolean networks

We examine complete synchronization of two deterministic Boolean networks (BNs) coupled unidirectionally in the drive-response configuration. A necessary and sufficient criterion is presented in terms of algebraic representations of BNs. As a consequence, we show that complete synchronization can occur only between two conditionally identical BNs when the transition matrix of the drive network is nonsingular. Two examples are worked out to illustrate the obtained results.

On some necessary and sufficient conditions for a recurrent neural network model with time delays to generate oscillations

In this paper, the existence of oscillations for a class of recurrent neural networks with time delays between neural interconnections is investigated. By using the fixed point theory and Liapunov functional, we prove that a recurrent neural network might have a unique equilibrium point which is unstable. This particular type of instability, combined with the boundedness of the solutions of the system, will force the network to generate a permanent oscillation. Some necessary and sufficient conditions for these oscillations are obtained. Simple and practical criteria for fixing the range of parameters in this network are also derived. Typical simulation examples are presented.

Analyzing inner and outer synchronization between two coupled discrete-time networks with time delays

This paper studies two kinds of synchronization between two discrete-time networks with time delays, including inner synchronization within each network and outer synchronization between two networks. Based on Lyapunov stability theory and linear matrix inequality (LMI), sufficient conditions for two discrete-time networks to be asymptotic stability are derived in terms of LMI. Finally numerical examples are given to illustrate the effectiveness of our derived results. The theoretical understanding provides insights into the dynamics of two or more neural networks with appropriate couplings.

Computation of synchronized periodic solution in a BAM network with two delays

A bidirectional associative memory (BAM) neural network with four neurons and two discrete delays is considered to represent an analytical method, namely, perturbation-incremental scheme (PIS). The expressions for the periodic solutions derived from Hopf bifurcation are given by using the PIS. The result shows that the PIS has higher accuracy than the center manifold reduction (CMR) with normal form for the values of time delay not far away from the Hopf bifurcation point. In terms of the PIS, the necessary and sufficient conditions of synchronized periodic solution arising from a Hopf bifurcation are obtained and the synchronized periodic solution is expressed in an analytical form. It can be seen that theoretical analysis is in good agreement with numerical simulation. It implies that the provided method is valid and the obtained result is correct. To the best of our knowledge, the paper is the first one to introduce the PIS to study the periodic solution derived from Hopf bifurcation for a 4-D delayed system quantitatively.

Globally exponential synchronization and synchronizability for general dynamical networks

The globally exponential synchronization problem for general dynamical networks is considered in this paper. One quantity will be distilled from the coupling matrix to characterize the synchronizability of the corresponding dynamical networks. The calculation of such a quantity is very convenient even for large-scale networks. The network topology is assumed to be directed and weakly connected, which implies that the coupling configuration matrix can be asymmetric, weighted, or reducible. This assumption is more consistent with the realistic network in practice than the constraint of symmetry and irreducibility. By using the Lyapunov functional method and the Kronecker product techniques, some criteria are obtained to guarantee the globally exponential synchronization of general dynamical networks. In addition, numerical examples, including small-world and scale-free networks, are given to demonstrate the theoretical results. It will be shown that our criteria are available for large-scale dynamical networks.