Multiple‐interval‐dependent robust stability analysis for uncertain stochastic neural networks with mixed‐delays

This article deals with the problem of robust stochastic asymptotic stability for a class of uncertain stochastic neural networks with distributed delay and multiple time‐varying delays. It is noted that the reciprocally convex approach has been intensively used in stability analysis for time‐delay systems in the past few years. We will extend the approach from deterministic time‐delay systems to stochastic time‐delay systems. And based on the new technique dealing with matrix cross‐product and multiple‐interval‐dependent Lyapunov–Krasovskii functional, some novel delay‐dependent stability criteria with less conservatism and less decision variables for the addressed system are derived in terms of linear matrix inequalities. At last, several numerical examples are given to show the effectiveness of the results. © 2014 Wiley Periodicals, Inc. Complexity 21: 147–162, 2015     

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.     

Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system.     

Cahn-Hilliard方程非条件能量稳定二阶方法的稳定性和收敛性分析

在这项工作中,我们构建了一种非条件能量稳定的高效不变能量四分法(IEQ)来求解Cahn-Hilliard方程.所构建的数值格式是线性的、具有二阶时间精度和非条件能量稳定性.我们仔细分析了数值格式的唯一可解性、稳定性和误差估计.结果表明,所构建的格式满足唯一可解性、非条件能量稳定性和时间方向的二阶收敛性.通过大量的二维和三维数值实验,我们进一步验证了所提出格式的收敛阶、非条件能量稳定性和有效性.     

带非线性源项的Riesz回火分数阶扩散方程的预估校正方法

本文针对带非线性源项的Riesz回火分数阶扩散方程,利用预估校正方法离散时间偏导数,并用修正的二阶Lubich回火差分算子逼近Riesz空间回火的分数阶偏导数,构造出一类新的数值格式.给出了数值格式在一定条件下的稳定性与收敛性分析,且该格式的时间与空间收敛阶均为二阶.数值试验表明数值方法是有效的.     

基于LMI的时滞细胞神经网络的全局渐近稳定性分析

研究了一类具有时滞的细胞神经网络的稳定性问题,利用Liapunov-Krasovskii泛函的方法,给出了时滞相关的稳定性判据．稳定性判据是以线性矩阵不等式(LMI)的形式给出,可以很容易得出时滞的上界．在得到时滞相关的稳定性判据的同时也可以得到时滞无关的稳定性判据,包含了已有文章中的很多结果．最后,数值算例说明了结果的优越性．     

Conserved quantities of some Hamiltonian wave equations after full discretization

Hamiltonian PDEs have some invariant quantities, which would be good to conserve with the numerical integration. In this paper, we concentrate on the nonlinear wave and Schrödinger equations. Under hypotheses of regularity and periodicity, we study how a symmetric space discretization makes that the space discretized system also has some invariants or `nearly' invariants which well approximate the continuous ones. We conjecture some facts which would explain the good numerical approximation of them after time integration when using symplectic Runge-Kutta methods or symmetric linear multistep methods for second-order systems.     

Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.     

Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations

Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of stability regions defined by using scalar test equations. The analysis is closely related to the stability analysis of the standard linear multistep methods for delay differential equations. A new second-order IMEX method which has approximately the same stability region as that of the IMEX Euler method, the simplest IMEX method of order 1, is proposed. Some numerical results are also presented which show superiority of the new method.      

Convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay

In this paper, we study the convergence and stability of the stochastic theta method (STM) for a class of index 1 stochastic delay differential algebraic equations. First, in the case of constrained mesh, i.e., the stepsize is a submultiple of the delay, it is proved that the method is strongly consistent and convergent with order 1/2 in the mean-square sense. Then, the result is further extended to the case of non-constrained mesh where we employ linear interpolation to approximate the delay argument. Later, under a sufficient condition for mean-square stability of the analytical solution, it is proved that, when the stepsizes are sufficiently small, the STM approximations reproduce the stability of the analytical solution. Finally, some numerical experiments are presented to illustrate the theoretical findings.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

Efficient unconditionally stable numerical schemes for a modified phase field crystal model with a strong nonlinear vacancy potential

We consider numerical approximations for a modified phase field crystal model with a strong nonlinear vacancy potential. Based on the invariant energy quadratization approach and stabilized strategies, we develop linear, unconditionally energy stable numerical schemes using the first-order Euler method, the second-order backward differentiation formulas and the second-order Crank–Nicolson method, respectively. We rigorously prove the unconditional energy stability, the mass conservation of these three numerical schemes and carry out error estimates in time for the first-order numerical scheme. Various numerical experiments in 2D and 3D are carried out to validate the accuracy, energy stability, mass conservation, and efficiency of the proposed schemes.     

Delay decomposition approach to state estimation of neural networks with mixed time‐varying delays and Markovian jumping parameters

This paper investigates the state estimation of neural networks with mixed time‐varying delays and Markovian jumping parameters. By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration. On the basis of the new Lyapunov–Krasovskii functional, some inequality techniques, stochastic stability theory and delay‐dependent stability criteria are obtained in terms of linear matrix inequalities. Finally, three numerical examples are given to illustrate the less conservative and effectiveness of our theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

解二阶抛物型方程组的再生核函数法

建立了二阶抛物型方程组的一种新数值方法-再生核函数法.利用再生核函数,直接给出了每个离散时间层上近似解的显式表达式,由显式表达式可实现完全并行计算;用能量估计法证明了格式的稳定性及二阶收敛性;给出了一些数值结果.     

Exponential stability for markovian jumping stochastic BAM neural networks with mode‐dependent probabilistic time‐varying delays and impulse control

In this article, an exponential stability analysis of Markovian jumping stochastic bidirectional associative memory (BAM) neural networks with mode‐dependent probabilistic time‐varying delays and impulsive control is investigated. By establishment of a stochastic variable with Bernoulli distribution, the information of probabilistic time‐varying delay is considered and transformed into one with deterministic time‐varying delay and stochastic parameters. By fully taking the inherent characteristic of such kind of stochastic BAM neural networks into account, a novel Lyapunov‐Krasovskii functional is constructed with as many as possible positive definite matrices which depends on the system mode and a triple‐integral term is introduced for deriving the delay‐dependent stability conditions. Furthermore, mode‐dependent mean square exponential stability criteria are derived by constructing a new Lyapunov‐Krasovskii functional with modes in the integral terms and using some stochastic analysis techniques. The criteria are formulated in terms of a set of linear matrix inequalities, which can be checked efficiently by use of some standard numerical packages. Finally, numerical examples and its simulations are given to demonstrate the usefulness and effectiveness of the proposed results. © 2014 Wiley Periodicals, Inc. Complexity 20: 39–65, 2015     

Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.     

Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay

In this paper we study a kind of second-order impulsive stochastic differential equations with state-dependent delay in a real separable Hilbert space. Some sufficient conditions for the approximate controllability of this system are formulated and proved under the assumption that the corresponding deterministic linear system is approximately controllable. The results concerning the existence and approximate controllability of mild solutions have been addressed by using strongly continuous cosine families of operators and the contraction mapping principle. At last, an example is given to illustrate the theory.     

Numerical algorithms for the time-Caputo and space-Riesz fractional Bloch-Torrey equations

In this paper, high-order numerical methods for time-Caputo and space-Riesz fractional Bloch-Torrey equations in one- and two-dimensional space are constructed, where the second-order backward fractional difference operator and the sixth-order fractional-compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results.     

Numerical solution of Burgers-Fisher equation by cubic B-spline quasi-interpolation

In this paper, numerical solution of Burgers-Fisher equation is presented based on the cubic B-spline quasi-interpolation. At first, the generalized Burgers-Fisher equation and the cubic B-spline quasi-interpolation are introduced. Moreover, the numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Moreover, the stability of this method is studied. At last, the numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method. The main advantage of the resulting scheme is that the algorithm is very simple, so it is very easy to implement.