A universal dichotomy for dynamical systems with variable delay

We show that the dynamics of systems with time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating to the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with constant delay. The new, second class is characterized by modelocking behavior of their access maps and by an asymptotically linear, instead of a logarithmic scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.