Method Of Differential Inequalities Research Articles

Pinning synchronization of two general complex networks with periodically intermittent control

AbstractIn this paper, the method of periodically pinning intermittent control is introduced to solve the problem of outer synchronization between two complex networks. Based on the Lyapunov stability theory, differential inequality method and adaptive technique, some simple synchronous criteria have been derived analytically. At last, both the theoretical and numerical analysis illustrate the effectiveness of the proposed control methodology. This method not only reduces the conservatism of control gain but also saves the cost of production.These advantages make this method having a large application scope in the real production process.

Finite-time outer synchronization between two complex dynamical networks with on–off coupling

In this paper, the finite-time outer synchronization between two complex dynamical networks with on–off coupling is investigated. By using suitable on–off controllers, sufficient conditions for finite-time outer synchronization are derived based on the Lyapunov function and the finite-time differential inequality method. The theoretical results imply that the two networks will achieve finite-time outer synchronization for fixed on–off rate if the control coupling strength is large enough. Numerical simulations are given to demonstrate the effectiveness of the theoretical results.

Existence and stability of periodic contrast structures in the reaction-advection-diffusion problem

A singularly perturbed periodic problem for a parabolic reaction-advection-diffusion equation at low advection is studied. The case when there is an internal transition layer under unbalanced nonlinearity is considered. An asymptotic expansion of a solution is constructed. To substantiate the asymptotics thus constructed, the asymptotic method of differential inequalities is used. The Lyapunov asymptotic stability of a periodic solution is studied; the proof uses the Krein-Rutman theorem.

Asymptotics of the Solution of the Bisingular Problem for a System of Linear Parabolic Equations. II

Suppose we are given a bisingular initial boundary-value problem for a system of parabolic equations that contains a small parameter ε ² at the second derivative and √ ε at the first derivative with respect to the spatial variable. We prove an asymptotics of any order for the solution of the problem with respect to the small parameter, without using the joining of asymptotic expansions. To this end, we apply an asymptotic method of differential inequalities. The essence of the method is to use the formal asymptotics (given in the previous paper) for constructing lower and upper solutions of the problem. By modifying the last terms of order ε ⁿ⁄² in the partial sum of the formal asymptotics, we construct the lower and the upper solutions, between which the exact solution of the problem lies.

Oscillation of certain nonlinear fractional partial differential equation with damping term

In this paper, we investigate the oscillatory behavior of solutions of the nonlinear fractional partial differential equation with damping and forced term subject to Robin boundary condition by using differential inequality method as well as integral average method. The main results are illustrated by examples.

Asymptotics of the front motion in the reaction-diffusion-advection problem

A singularly perturbed initial boundary value problem is considered for a parabolic equation that is known in application as the reaction-diffusion-advection equation. An asymptotic expansion of solutions with a moving front is constructed. This asymptotics is proved by the method of differential inequalities, which is based on well-known comparison theorems and develops the ideas of formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.

Functional differential inequalities with partial derivatives

Initial problems for Hamilton--Jacobi functional differential equations and initial boundary value problems of the Dirichlet type for parabolic equations are considered. It is proved that classical solutions of functional differential equations can be estimated by solutions of initial problems for ordinary functional differential equations. Theorems on the uniqueness of solutions are obtained as consequences of comparison results. A method of differential inequalities is used. Here, the involved operators do not satisfy the Volterra condition.

A steplike contrast structure in a singularly perturbed system of elliptic equations

A singularly perturbed boundary value problem for a system of elliptic equations in a two-dimensional region is considered. The asymptotics and existence of a solution with an internal transition layer are studied. The asymptotics is justified by the method of differential inequalities.

Studying the intermittent stable theorem and the synchronization of a delayed fractional nonlinear system

In this paper, an intermittent synchronizing delayed fractional nonlinear system is studied. We propose a novel intermittent stable theorem for the delayed fractional system and derive a new synchronization criterion for delayed fractional systems by means of fractional stable theorem and the differential inequality method. Intermittent synchronizing fractional delayed Newton—Leipnik system is taken as an illustrative example and numerical simulation of this example is presented to show the feasibility and effectiveness of the proposed theorem.

Exponential Stabilization and Synchronization of Complex-variable Chaotic Systems via Intermittent Control

Abstract In this paper, exponential stabilization and synchronization of complex-variable chaotic system are investigated. Periodically intermittent controllers are designed and applied to achieve the stabilization and synchronization. Based on the differential inequality method and the analysis technique, several simple and useful synchronization criteria are derived. Simulated examples are provided to verify the effectiveness of the proposed method.

Contrast structures in singularly perturbed quasilinear reaction-diffusion-advection equations

We consider stationary solutions with internal transition layers (contrast structures) for a singularly perturbed elliptic equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an efficient algorithm for constructing an asymptotic approximation to the localization curve of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions.

Contrast structures in the reaction-diffusion-advection equations in the case of balanced advection

For a singularly perturbed parabolic equation termed in applications as the reaction-diffusion-advection equation, stationary solutions with internal transition layers (contrast structures) are studied. An arbitrary-order asymptotic approximation of such solutions is constructed, and an existence theorem is proved. An efficient algorithm for constructing an asymptotic approximation of the transition point is proposed. The constructed asymptotic approximation is justified by applying the asymptotic method of differential inequalities, which is extended to the class of problems under study. This method is also used to establish the Lyapunov stability of such stationary solutions.

Steplike contrast structure in a singularly perturbed system of equations with different powers of small parameter

A singularly perturbed boundary value problem for a system of equations with different powers of a small parameter is considered in the one-dimensional case. The asymptotic behavior and existence of a solution with an internal transition layer are analyzed. The asymptotics are substantiated using the asymptotic method of differential inequalities.

The initial boundary value problem for a nonlocal singularly perturbed reaction-diffusion equation

The initial boundary value problem for a nonlinear singularly perturbed integro-parabolic equation is examined. An asymptotic expansion of the solution to this problem containing the temporal, spatial and corner boundary layers is constructed. The existence and local uniqueness of the solution is justified by using the asymptotic method of differential inequalities.

Dynamical analysis of impulsive neural networks with time‐varying delays

In this work, a new criterion concerning the global exponential stability of impulsive neural networks with time‐varying delays is presented by employing the impulsive delayed differential inequality method. The criterion is independent of the time‐varying delays and does not require the differentiability of delay functions. An example and its simulation showing the effectiveness of the present criterion is given finally. Copyright © 2012 John Wiley & Sons, Ltd.

Contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems

We consider a nonlinear boundary value problem for an equation of reaction-diffusion-advection type whose solutions have internal transition layers (contrast structures). We prove the existence of such solutions, construct their asymptotics, and estimate its accuracy. The analysis is carried out with the use of the asymptotic method of differential inequalities and permits one to prove the Lyapunov stability of the contrast structures.

Delayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed boundary value problem with a turning point

Based on the method of differential inequalities, by constructing the upper ad lower solutions suitably, delayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed Dirichlet boundary value problem with a turning point is found in this paper. An illustrating example is performed to verify the obtained results.

Moving fronts in integro-parabolic reaction-advection-diffusion equations

We consider initial-boundary value problems for a class of singularly perturbed nonlinear integro-differential equations. In applications, they are referred to as nonlocal reactionadvection-diffusion equations, and their solutions have moving interior transition layers (fronts). We construct the asymptotics of such solutions with respect to a small parameter and estimate the accuracy of the asymptotics. To justify the asymptotics, we use the asymptotic differential inequality method.

SYNCHRONIZATION OF COMPLEX NETWORK WITH NON-DELAYED AND DELAYED COUPLING VIA INTERMITTENT CONTROL

This paper investigates the exponential synchronization of complex network with non-delayed and delayed coupling. A periodically intermittent controller is designed to synchronize the network onto a given orbit. Based on Lyapunov function and differential inequality method, the criteria for exponential synchronization are derived. Numerical simulation is presented to verify the effectiveness of the derived results.

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.