  We show that delay-differential equations (DDE) exhibit universal bifurcation
scenarios, which are observed in large classes of DDEs with a single delay.
Each such universality class has the same sequence of stabilizing or
destabilizing Hopf bifurcations. These bifurcation sequences and universality
classes can be explicitly described by using the asymptotic continuous spectrum
for DDEs with large delays. Here, we mainly study linear DDEs, provide a
general transversality result for the delay-induced bifurcations, and consider
three most common universality classes. For each of them, we explicitly
describe the sequence of stabilizing and destabilizing bifurcations. We also
illustrate the implications for a nonlinear Stuart-Landau oscillator with
time-delayed feedback.
