Delayed Negative Feedback: A Warm-Up

Abstarct

A great deal about delay differential equations can be learned by a study of its simplest representative, the linear delayed negative feedback equation.We use it to illustrate features common to delay differential equations, such as the tendency of delays to give rise to oscillations that can become undamped if delays are large.The obstructions to solving delay differential equations backwards in time are readily appreciated for this simple equation. It is an unpleasant fact that even for this simple linear equation, the stability of the trivial equilibrium requires an analysis of the roots of a transcendental equation. We show how the leading root of this transcendental equation signals the oscillation in solutions of the delay differential equation.