Disc theorem

Abstract: Global chaos synchronization of two identical nonlinear transducer systems is investigated via linear state error feedback control. The sufficient criterion for global chaos synchronization is derived firstly by the Gerschgorin disc theorem and the stability theory of linear time-varied systems. Then this sufficient criterion is further optimized in the sense of reducing the lower bounds of the coupling coefficients with two methods, one based on Gerschgorin disc theorem itself and the other based on Lyapunov direct method. Finally, two optimized criteria are compared theoretically.

Abstract: Two sufficient conditions for a generic master-slave synchronization scheme by linear state error feedback control are derived based respectively on Sylvester's criterion and Gerschgorin disc theorem, which are expressed by some inequalities. It is proven theoretically that the sufficient condition by Sylvester's criterion is more flexible than that by Gerschgorin disc theorem. The two sufficient conditions are further compared by three typical chaotic systems: the Duffing equation of two dimensions, the Lorenz system of three dimensions and a loudspeaker system of four dimensions. Numerical simulations support the theoretical result.

Abstract: The consensus algorithm with a static leader is proposed to solve the consensus problem of the discrete-time second-order multi-agent systems with communication delay.By the generalized Nyquist criterion and the Gerschgorin disc theorem,the sufficient conditions are obtained for the system to converge to the leader’s states asymptotically.With the interconnection topology composed of the agents and the leader that satisfies certain connectivity conditions,the sufficient conditions are decentralized,dependent on the control parameters and the weights between the neighboring agents,and independent of the communication delay.Numerical examples are provided to illustrate the correctness of the results.

Abstract: This paper considers stability analysis of a discrete-time computer virus model in networks. The disease-free equilibrium and the disease equilibrium are first derived from the mathematical model. Then the sufficient condition of stability for the disease-free equilibrium is obtained by the first Lyapunov method. And the sufficient conditions of stability for the disease equilibrium are given by disc theorem. Simulation results demonstrate the effectiveness of the stability conditions.