Approaches toward Understanding Delay-Induced Stability and Instability

For students in applied mathematics courses, the phenomenon of delay-induced stability and instability offers exciting educational opportunities. Exploration of the onset of instability in delay differential equations (DDEs) invites a blend of analysis (real, complex, and functional), algebra, and computational methods. Moreover, stabilization of unstable but "desirable" equilibria using delayed feedback is of high importance in science and engineering. The primary challenge in classifying the stability of equilibria of DDEs lies in the fact that characteristic equations are transcendental. Here, we survey two approaches for understanding the stability of equilibria of DDEs. The first approach uses a functional analytic framework, a departure from the more familiar textbook methods based upon characteristic equations and completeness-type arguments. The second approach uses Pontryagin's generalization of the Routh-Hurwitz conditions. We apply the latter approach to illustrate how the deliberate introduction of a second time delay in a single-delay differential equation can stabilize an otherwise unstable equilibrium.