Wenlian Lu

Pinning Complex Networks by a Single Controller

In this paper, without assuming symmetry, irreducibility, or linearity of the couplings, we prove that a single controller can pin a coupled complex network to a homogenous solution. Sufficient conditions are presented to guarantee the convergence of the pinning process locally and globally. An effective approach to adapt the coupling strength is proposed. Several numerical simulations are given to verify our theoretical analysis.

Synchronization of coupled connected neural networks with delays

We investigate synchronization of an array of linearly coupled identical connected neural networks with delays; Variational method is used to investigate local synchronization. Global exponential stability is studied, too. We do not assume that the coupling matrix A is symmetric or irreducible. The linear matrix inequality approach is used to judge synchronization with global convergence property.

Cluster synchronization in networks of coupled nonidentical dynamical systems

In this paper, we study cluster synchronization in networks of coupled nonidentical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two conditions play key roles for cluster synchronization: the common intercluster coupling condition and the intracluster communication. From the latter one, we interpret the two cluster synchronization schemes by whether the edges of communication paths lie in inter- or intracluster. By this way, we classify clusters according to whether the communications between pairs of vertices in the same cluster still hold if the set of edges inter- or intracluster edges is removed. Also, we propose adaptive feedback algorithms to adapting the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the conditions of common intercluster coupling and intracluster communication. We also give several numerical examples to illustrate the theoretical results.

Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions

In this paper, we discuss dynamics of Cohen-Grossberg neural networks with discontinuous activations functions. We provide a relax set of sufficient conditions based on the concept of Lyapunov diagonally stability (LDS) for Cohen-Grossberg networks to be absolutely stable. Moreover, under certain conditions we prove that the system is exponentially stable globally or convergent globally in finite time. Convergence rate for global exponential convergence and convergence time for global convergence in finite time are also provided.

New conditions on global stability of Cohen-Grossberg neural networks

In this letter, we discuss the dynamics of the Cohen-Grossberg neural networks. We provide a new and relaxed set of sufficient conditions for the Cohen-Grossberg networks to be absolutely stable and exponentially stable globally. We also provide an estimate of the rate of convergence.

Consensus of Multi-Agent Systems With Unbounded Time-Varying Delays

In this note, the consensus problem with infinite time-varying delays for linearly coupled static network is investigated. The delay affects only the off-diagonal terms in continuous-time equations. At first, we define an effective consensus ability index. Then, by using the graph theory and a new concept of consensus, we prove that under some mild conditions, the network can realize consensus. An example is given to show the validity of obtained results.

Neural networks letter: Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches

In this paper, global dissipativity and quasi-synchronization issues are investigated for the delayed neural networks with discontinuous activation functions. Under the framework of Filippov solutions, the existence and dissipativity of solutions can be guaranteed by the matrix measure approach and the new obtained generalized Halanay inequalities. Then, for the discontinuous master-response systems with parameter mismatches, quasi-synchronization criteria are obtained by using feedback control. Furthermore, when the proper approximate functions are selected, the complete synchronization can be discussed as a special case that two systems are identical. Numerical simulations on the chaotic systems are presented to demonstrate the effectiveness of the theoretical results.

Dynamical Behaviors of Delayed Neural Network Systems with Discontinuous Activation Functions

In this letter, without assuming the boundedness of the activation functions, we discuss the dynamics of a class of delayed neural networks with discontinuous activation functions. A relaxed set of sufficient conditions is derived, guaranteeing the existence, uniqueness, and global stability of the equilibrium point. Convergence behaviors for both state and output are discussed. The constraints imposed on the feedback matrix are independent of the delay parameter and can be validated by the linear matrix inequality technique. We also prove that the solution of delayed neural networks with discontinuous activation functions can be regarded as a limit of the solutions of delayed neural networks with high-slope continuous activation functions.

Almost periodic dynamics of a class of delayed neural networks with discontinuous activations

We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.

Coexistence and local stability of multiple equilibria in neural networks with piecewise linear nondecreasing activation functions

In this paper, we investigate the neural networks with a class of nondecreasing piecewise linear activation functions with 2r corner points. It is proposed that the n-neuron dynamical systems can have and only have (2r+1)(n) equilibria under some conditions, of which (r+1)(n) are locally exponentially stable and others are unstable. Furthermore, the attraction basins of these stationary equilibria are estimated. In the case of n=2, the precise attraction basin of each stable equilibrium point can be figured out, and their boundaries are composed of the stable manifolds of unstable equilibrium points. Simulations are also provided to illustrate the effectiveness of our results.