Asymptotic dynamics of ordinary differential equations (ODEs) are commonly understood by looking at eigenvalues of a matrix, and transient dynamics can be bounded above and below by considering the corresponding pseudospectra. While asymptotics for other classes of differential equations have been studied using eigenvalues of a (nonlinear) matrix-valued function, there are no analogous pseudosp...

Consider the first-order linear delay differential equation x′(t) + p(t)x(τ(t)) = 0, t ≥ t0, and the second-order linear delay equation x′′(t) + p(t)x(τ(t)) = 0, t ≥ t0, where p and τ are continuous functions on [t0,∞), p(t) > 0, τ(t) is nondecreasing, τ(t) ≤ t for t ≥ t0 and limt→∞ τ(t) = ∞. Several oscillation criteria are presented for the first-order equation when

In this paper, a higher order nonlinear neutral functional differential equation with distributed delay is studied by using the continuation theorem of coincidence degree theory. Some new results on the existence of periodic solutions are obtained. Keywords—Neutral functional differential equation, higher order, periodic solution, coincidence degree theory

In this paper, by considering ordinary differential equation (ODE) as a special case and a starting point of delay differential equation (DDE), we will show that some typical topological methods such as continuation theorems can be applied to detect some dynamics of DDE like periodic solutions. Several problems will be presented.

We consider two identical oscillators with time delayed coupling, modelled by a system of delay differential equations. We reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. By analyzing the phase model, we show how the time delay affects the stability of phase-locked periodic solutions and causes stability switching of in-phase and ...

Neutral differential equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines; see Hale 1 . In recent years, many studies have been made on the oscillatory behavior of solutions of neutral delay differential equations, and we refer to the recent papers 2–23 and the ...

where x t ( ) x (t ) and 0. Observe that xt ( ) with 0 represents a portion of the solution trajectory in a recent past. Here, f is a functional operator that takes a time input and a continuous function xt ( ) with 0 and generates a real number (dx (t )/dt ) as its output. A well-known example of a delay differential equation is the Hutchinson equation, or the discrete delay logistic equation,...

We prove that a function of several variables satisfying a functional differential inequality with unbounded delay can be estimated by a solution of a suitable initial problem for an ordinary functional differential equation. As a consequence of the comparison theorem we obtain a Perrontype uniqueness result and a result on continuous dependence of solutions on given functions for partial funct...

In many fields of the contemporary science and technology, systems with delaying links often appear. By a delay differential equation (DDE), we mean an evolutionary system in which the (current) rate of change of the state depends on the historical status of the system. Delay models play a relevant role in different fields such as biology, economy, control, and electrodynamics and hence have be...

By using the upperand lower-solution method of partial functional differential equations and the oscillation theory of functional differential equation, the oscillation of a population equation with diffusion and delay is studied and a sufficient condition for all positive solutions of the equation to oscillate about the positive equilibrium is obtained. Finally, a model arising from ecology is...