Xiaohua Ding

Nonstandard finite difference variational integrators for nonlinear Schrödinger equation with variable coefficients

In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These integrators are naturally multi-symplectic, and their multi-symplectic structures are presented by the multi-symplectic form formulas. Local truncation errors and convergences of the integrators are briefly discussed. The effectiveness and efficiency of the proposed schemes, such as the convergence order, numerical stability, and the capability in preserving the norm conservation, are verified in the numerical experiments.

Global dissipativity of a class of BAM neural networks with both time-varying and continuously distributed delays

In this paper, the global dissipativity of a class of BAM neural networks with both time-varying and continuously distributed delays is investigated. By constructing Lyapunov functions and employing a generalized Halanay inequality and the linear matrix inequality (LMI), several sufficient easy-to-test conditions for the global dissipativity of the underling system are successfully derived. The results extend and improve some previous publications on conventional BAM neural networks. Moreover, the estimations of the positive invariant set, globally attractive set and globally exponential attractive set are carried out. Finally, examples are given to demonstrate the effectiveness of the obtained results.

Exact Finite Difference Scheme and Nonstandard Finite Difference Scheme for Burgers and Burgers-Fisher Equations

We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes.

Exact finite-difference scheme and nonstandard finite-difference scheme for coupled Burgers equation

This work develops exact finite-difference schemes for the two-dimensional nonlinear coupled viscous Burgers equation using the analytic solution. We extend the explicit nonstandard finite-difference schemes on the basis of the exact finite-difference schemes to solve the coupled Burgers equation. Numerical examples are presented to verify the efficiency and accuracy of the methods.MSC:39A10, 65L12, 74H15.

Impulsive stabilization of delay difference equations and its application in Nicholson's blowflies model

In this article, we consider the impulsive stabilization of delay difference equations. By employing the Lyapunov function and Razumikhin technique, we establish the criteria of exponential stability for impulsive delay difference equations. As an application, by using the results we obtained, we deal with the exponential stability of discrete impulsive delay Nicholsons blowflies model. At last, an example is given to illustrate the efficiency of our results.Mathematics Subject Classification 2000: 39A30; 39A60; 39A10; 92B05.

Razumikhin method conjoined with graph theory to input-to-state stability of coupled retarded systems on networks

Abstract This paper is concerned with input-to-state stability (ISS) of coupled retarded systems on networks (CRSNs). By incorporating Razumikhin method with generalized Kirchhoff’s Matrix Tree Theorem, sufficient conditions are established to guarantee ISS of CRSNs. For the convenience of application, sufficient conditions in the form of coefficients are also obtained. What is more, a sufficient criterion on ISS of the coupled retarded oscillators system on networks is derived. Finally, as an illustration, a numerical example is given to demonstrate the effectiveness of the theoretical results.

The existence of periodic solutions for coupled Rayleigh system

Abstract This paper is concerned with the existence of periodic solutions for coupled Rayleigh system (CRS). A sufficient criterion for the existence of periodic solutions for CRS is provided via an innovative method of combining graph theory with coincidence degree theory as well as Lyapunov method. As a subsequent result, coupled Lord Rayleigh system is also discussed. Subsequently, a sufficient condition is given to determine the existence of its periodic solutions. Finally, a numerical example and its simulations are presented to illustrate the effectiveness and feasibility of the proposed criterion.

On the Difference Equation

We study a discrete delay Mosquito population equation. Firstly, we study the stability of the equilibria of the system and the existence of period-two bifurcation by analyzing the characteristic equation. Secondly, the direction and stability of the bifurcation are determined by using the normal form theory. Finally, some computer simulations are performed to illustrate the analytical results found.

Global asymptotic stability of a class of generalized BAM neural networks with reaction-diffusion terms and mixed time delays

Abstract In this paper, a novel linear matrix inequality (LMI)-based sufficient condition, which guarantees the existence and global asymptotic stability of a class of generalized bidirectional associative memory (BAM) neural networks with reaction-diffusion terms and mixed time delays, is obtained by using inequality technique, degree theory, LMI method and constructing Lyapunov functional. The mixed time delays consist of both the discrete delays and the infinitely distributed delays. The results generalize and improve the earlier publications under the assumption that the activation functions only satisfy general global Lipschitz conditions. Two simple examples are provided to demonstrate the effectiveness of the proposed theoretical results. These results can be applied to design globally asymptotically stable networks and thus have important significance in both theory and applications.