To study the nonlinear dynamics, such as Hopf bifurcation, of partial differential equations with delay, one needs to consider the characteristic equation associated to the linearized equation and to determine the distribution of the eigenvalues; that is, to study the spectrum of the linear operator. In this paper we study the projectors on the generalized eigenspaces associated to some eigenva...

In this paper, we study the instability of the zero solution to a nonlinear differential equation with variable delay. By using the Lyapunov functional approach, some sufficient conditions for instability of the zero solution are obtained. Keywords—Instability, Lyapunov-Krasovskii functional, delay differential equation, fifth order.

A partial differential equation with the argument [Z.t] is studied, where [.] denotes the greatest integer function. The infinite delay -[Zt] leads to difference equations of unbounded order. KEY WORDSAND PHRASES. Partial differential equation, piecewise constant delay, boundary value problem, initial value problem. 1991 AMS SUBJECT SIFICATION CODE. 35A05, 35B25, 35L10, 34K25.

Consider the delay differential equation (1) ẋ(t) = g(x(t), x(t − r)), where r > 0 is a constant and g : 2 → is Lipschitzian. It is shown that if r is small, then the solutions of (1) have the same convergence properties as the solutions of the ordinary differential equation obtained from (1) by ignoring the delay.

It is demonstrated that the method of steps for linear delay-differential equation together with the inverse Laplace transform can be used to find a converging sequence of polynomial approximants to the transcendental function determining stability of the delay equation. Numerical stability charts are shown to illustrate convergence. This approach can serve as a basis for an efficient numerical...

We are concerned with the exponential mean-square stability of two-step Maruyama methods for stochastic differential equations with time delay. We propose a family of schemes and prove that it can maintain the exponential mean-square stability of the linear stochastic delay differential equation for every step size of integral fraction of the delay in the equation. Numerical results for linear ...

The current final project belongs to a subject in the master’s degree in Advanced Mathematics at the University of Barcelona. It deals with time delays which usually are arisen in differential equations. Firstly, the project develops the main important known results of Delay Differential Equations, which are a specific case of Functional Differential Equations. In particular, we shall also focu...

Uncertain delay differential equation (UDDE) is a type of differential equations driven by Liu process. It has been proved that uncertain delay differential equation has a unique solution in the finite domain, under the conditions that the coefficients are global Lipschitz continuous. This paper will extend this existence and uniqueness theorem from finite domain to infinite domain under the lo...

In this work, we successfully extended one-dimensional differential transform method (DTM), by presenting and proving some theorems, to solving nonlinear high-order multi-pantograph equations. This technique provides a sequence of functions which converges to the exact solution of the problem. Some examples are given to demonstrate the validity and applicability of the present method and a comp...

Chaos synchronization of coupled fractional order differential equation is receiving increasing attention because of its potential applications in secure communications and control processing. The aim of this paper is synchronization between two identical or different delay fractional-order chaotic systems in finite time. At first, the predictor-corrector method is used to obtain the solutions ...