Title: From dynamical systems with time-varying delay to circle maps and Koopmanism,
Abstract: In the present paper we investigate the influence of the retarded access by a
time-varying delay on the dynamics of delay systems. We show that there are two
universality classes of delays, which lead to fundamental differences in
dynamical quantities such as the Lyapunov spectrum. Therefore we introduce an
operator theoretic framework, where the solution operator of the delay system
is decomposed into the Koopman operator describing the delay access and an
operator similar to the solution operator known from systems with constant
delay. The Koopman operator corresponds to an iterated map, called access map,
which is defined by the iteration of the delayed argument of the delay
equation. The dynamics of this one-dimensional iterated map determines the
universality classes of the infinite-dimensional state dynamics governed by the
delay differential equation. In this way, we connect the theory of time-delay
systems with the theory of circle maps and the framework of the Koopman
operator. In the present paper we extend our previous work [Otto, Müller, and
Radons, Phys. Rev. Lett. 118, 044104 (2017)], by elaborating the mathematical
details and presenting further results also on the Lyapunov vectors.