In this paper we give an example of Markus-Yamabe instability in a constant coefficient delay differential equation with time-varying delay. For all values of the range of the delay function, the characteristic function of the associated autonomous delay equation is exponentially stable. Still, the fundamental solution of the time-varying system is unbounded. We also present a modified example ...

Uncertain delay differential equation is a type of differential equations driven by a canonical Liu process. This paper mainly focuses on the stability of uncertain delay differential equations. At first, the concept of stability in measure, stability in mean and stability in moment for uncertain delay differential equations will be presented. In addition, the sufficient condition for uncertain...

Models with a time delay often occur, since there is a naturally occurring delay in the transmission of information. A model with a delay can be noninvertible, which in turn leads to qualitative differences between the dynamical properties of a delay equation and the familiar case of an ordinary differential equation. We give specific conditions for the existence of noninvertible solutions in d...

In this article we have considered a stochastic delay differential equation model for two species competitive phytoplankton system with allelopathic stimulation. We have extended the deterministic model system to a stochastic delay differential equation model system by incorporating multiplicative white noise terms in the growth equations for both species. We have studied the mean square stabil...

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. Systemtheoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a sing...

A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condit...

The monodromy operator of a linear delay differential equation with periodic coefficients is formulated as an integral operator. The kernel of this operator includes a factor formed from the fundamental solution of the linear delay differential equation. Although the properties of the fundamental solutions are known, in general there is no closed form for the fundamental solution. This paper de...

In this paper, the following third-order nonlinear delay differential equation with periodic coefficients x′′′(t) + p(t)x′′(t) + q(t)x′(t) + r(t)x(t) = f (t, x (t) , x(t− τ(t))) + c(t)x′(t− τ(t)), is considered. By employing Green’s function and Krasnoselskii’s fixed point theorem, we state and prove the existence of positive periodic solutions to the third-order delay differential equation.

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often t...