Uncertain delayed genetic regulatory networks are investigated from an adaptive filtering approach based on an adaptive synchronization setting. For an unknown regulatory network with time delay and uncertain noise disturbance, several adaptive laws are derived to ensure the stochastic stability of the error states between the unknown network and the estimated model. The novelty lies in the fact that the designed adaptive laws are independent of the unknown system states and parameters, requiring only the output and structure of the underlying network. A representative simulation example is given to verify the effectiveness of the theoretical results.

This paper addresses the synchronized region problem, which is reduced to a matrix stability problem, for complex dynamical networks. For any natural number n, the existence of a network which has n disconnected synchronized regions is theoretically demonstrated. This shows the complexity in network synchronization. Convexity characteristic of stability for matrix pencils is further discussed. Smooth and generalized smooth Chua’s circuit networks are finally discussed as examples for illustration.

In this paper, the problem of pinning control for synchronization of complex dynamical networks is discussed. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is that lower cost is achieved by the control scheme of pinning nodes with smaller degrees. Some rigorous mathematical analysis is presented for achieving lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barab´asi-Albert model and various star-shaped networks are shown for verification and illustration.

This paper addresses the fundamental problem of complex network synchronizability from a graphtheoretic approach. First, the existing results are briefly reviewed. Then, the relationships between the network synchronizability and network structural parameters e.g., average distance, degree distribution, and node betweenness centrality are discussed. The effects of the complementary graph of a given network and some graph operations on the network synchronizability are discussed. A basic theory based on subgraphs and complementary graphs for estimating the network synchronizability is established. Several examples are given to show that adding new edges to a network can either increase or decrease the network synchronizability. To that end, some new results on the estimations of the synchronizability of coalescences are reported. Moreover, a necessary and sufficient condition for a network and its complementary network to have the same synchronizability is derived. Finally, some examples on Chua circuit networks are presented for illustration.

In this paper, the original Chua’s circuit is modified by substituting its piecewise-linear function with an attraction-repulsion function. Some new complex dynamical behaviors such as chaos are observed through computer simulations. Basic properties of the new circuit are analyzed by means of bifurcation diagrams. Lagrange stability conditions of the circuit are derived. A comparison between this modified Chua’s circuit with an attraction-repulsion function and the modified Chua’s circuit with a cubic nonlinear function is presented. Moreover, a generalization of the new circuit that can generate multiple scrolls is designed and simulated. Finally, a physical circuit is built to visualize the new system, with some experimental observations reported.