Global attractivity and oscillation in a nonlinear periodic delay difference equation

The existence of a periodic solution for a class of nonlinear delay difference equations with periodic coefficients is established. Some sufficient conditions for the global attractivity and oscillation about the periodic solutions are also obtained.     

具有变时滞阶段结构捕食系统的全局吸引性

研究了一类具有变时滞相互干扰阶段结构捕食系统。利用微分不等式比较原理和迭代法得到了系统平衡态全局吸引的充分条件,通过举例并进行数值模拟验证了结论的可行性。     

Stabilisation of second-order nonlinear equations with variable delay

For a wide class of second-order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal allows to stabilise unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilisation by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.     

Global satisfactory control for nonlinear integrator processes with long delay

Integrator processes with long delay are difficult to control. Nonlinear characteristics of actuators make the control problem more challenging. A technique is proposed in this paper for global satisfactory control （GSC） of such processes with relay-type nonlinearity. An oscillatory control signal is injected into the nonlinear process; the amplitude and frequency of the oscillatory signal are designed to linearise the nonlinear process in the sense of harmonic analysis; and a state feedback controller is configured to implement GSC over the linearised process. An illustrative example is given to demonstrate the effectiveness of     

Global satisfactory control for nonlinear integrator processes with long delay

Integrator processes with long delay are difficult to control. Nonlinear characteristics of actuators make the control problem more challenging. A technique is proposed in this paper for global satisfactory control (GSC) of such processes with relay-type nonlinearity. An oscillatory control signal is injected into the nonlinear process; the amplitude and frequency of the oscillatory signal are designed to linearise the nonlinear process in the sense of harmonic analysis; and a state feedback controller is configured to implement GSC over the linearised process. An illustrative example is given to demonstrate the effectiveness of the proposed method.     

Monotone methods for a finite difference system of reaction diffusion equation with time delay

The aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction-diffusion equations with time delays. The iterative schemes lead to computational algorithms as well as existence, uniqueness, and upper and lower bounds of the solution. An application to a diffusive logistic equation with time delay is given.     

Global stability for separable nonlinear delay differential equations

Consider the following separable nonlinear delay differential equation

, where we assume that, there is a strictly monotone increasing function f(x) on (−∞, +∞) such that

In this paper, to the above separable nonlinear delay differential equation, we establish conditions of global asymptotic stability for the zero solution. In particular, for a special wide class of f(x) which contains a case of f(x) = e^x−1, we give more explicit conditions. Applying these, we offer conditions of global asymptotic stability for solutions of nonautonomous logistic equations with delays.     

Global asymptotic stability of a nonlinear parabolic equation

Necessary and sufficient conditions are derived for the asymptotic stability of a nonlinear parabolic equation. Spatial Fourier transformation converts the stability determination to that for an infinite system of ordinary differential equations, and facilitates approximate to exact solutions, depending on the boundary conditions, the nonlinearities, and the harmonic content of the steady-state disturbances.     

Global attractivity in a competition system with feedback controls

Sufficient conditions are derived for the existence of a globally attracting positive equilibrium of a two species competition system with feedback controls; the indirect controls can act instantaneously or with a fixed discrete delay.     

Global stability analysis scheme for a class of nonlinear time delay systems

We consider a class of nonlinear time delay systems created by generalizing the model for FAST TCP, an Internet congestion control algorithm. We achieve a time delay independent sufficient condition for the global asymptotic stability of the class of systems. The sufficient condition is verified by constructing two sequences that represent the lower and upper bound variations of the system trajectory in time, and showing that the two sequences converge to the equilibrium point of the system. The simulation results exemplify that the sufficient condition is valid for global asymptotic stability, and that the sufficient condition is a close approximation to the unknown necessary and sufficient condition for global asymptotic stability.     

A finite difference method for korteweg-de vries like equation with nonlinear dispersion

The Korteweg-de Vries (KdV) equation has been generalized by Rosenau and Hyman [3] to a class of partial differential equations (PDEs) which has solitary wave solution with compact support. These solitary wave solutions are called compactons. Compactons are solitary waves with the remarkable soliton property, that after colliding with other compactons, they reemerge with the same coherent shape. These particle like waves exhibit elastic collision that are similar to the soliton interaction associated with completely integrable systems. The point where two compactons collide are marked by a creation of low amplitude compacton-anticompacton pair. These equations have only a finite number of local consevation laws. In this paper, an implicit numerical method has been developed to solve the K(2,3) equation. Accuracy and stability of the method have been studied. The analytical solution and the conserved quantities are used to assess the accuracy of the suggested method. The numerical results have shown that this compacton exhibits true soliton behavior.

     

Positive solutions of higher-order nonlinear delay difference equations

Classification schemes for positive solutions of a class of higher-order nonlinear delay difference equations are given in terms of their asymptotic behavior, and necessary as well as sufficient conditions for the existence of these solutions are also provided.     

General finite difference approximation for the wave equation with variable coefficients using a cubic spline technique

General implicit finite difference approximations for the wave equation with variable coefficients in one and two space variables are derived by using a cubic spline function and continuous parameter approach. An optimal scheme of the family and high accuracy schemes due to McKee are obtained for special cases of the parameters. The stability of the schemes are discussed and one test example is solved to illustrate the theory.     

Global stabilization of stochastic high-order feedforward nonlinear systems with time-varying delay

In this paper, we consider the problem of global stabilization for a class of stochastic high-order feedforward nonlinear systems with time-varying delay. By introducing the homogeneous domination design method and constructing the appropriate Lyapunov–Krasovskii functional, a state feedback controller is constructed to drive the closed-loop system to be globally asymptotically stable in probability.