I'm wondering if anyone can comment on the stability of delay DE given that we can analyze its characteristic equation.
For instance, let's say we have the DDE $\frac{d}{dt}x(t) = x(t-a),$ where $a$ is a constant.
Well, even though the history function can be literally any continuous function, we might, maybe, just maybe, might be able to say something concrete about the long-term stability by analyzing the eigen-equation $\lambda = \exp(-a \lambda).$
But is that true? How do you know? What credible references exist that prove you can analyze DDEs by looking at whether their eigenvalues have positive, negative and imaginary parts?
Thanks!
