Dynamics of Delay Equations, Theory and Applications - Abstract
Müller, David
We analyze general time-delay systems which arise in many fields such as life science, control theory, synchronization of networks, climate dynamics and engineering. In all of these fields time-varying delays are introduced for example to take into account environmental fluctuations, for improving control and synchronization strategies and lead to interesting dynamical properties. We identify two universality classes of delay differential equations with time-varying delay, where the classification depends only on the functional structure of the delay and therefore it is independent of the specific system. Systems with emphconservative delay are equivalent to systems with constant delay, where equivalent means that there exists a somehow well-behaved time scale transformation which transforms the original system to a system with constant delay. Hence the dynamics of systems with conservative delay equals the dynamics of systems with constant delay, which are extensively studied in the literature. Contrarily, systems with emphdissipative delay are not equivalent to systems with constant delay and the related dynamics differs from the dynamics of systems with constant delay. The classification we present is directly connected to the well-understood problem whether or not a specific circle map is topological conjugate to a pure rotation. In our case the circle map is defined by the delay and describes the dynamics of the repeated access of the delay system to the past. This connection leads to the interesting consequence that the type of a member of a typical parameter family of delays depends in a fractal manner on the parameters. To illustrate the consequences of the classification on the dynamics of general systems with time-varying delay, we present fundamental differences in the scaling behavior of the Lyapunov spectrum and the functional structure of the Lyapunov vectors between systems with conservative delay and systems with dissipative delay. In detail, we demonstrate that the scaling behavior of the Lyapunov spectrum is connected to the spectrum of a Koopman operator, which corresponds to the above-mentioned circle map. Systems with constant delay are dissipative systems due to the known logarithmic scaling of the Lyapunov spectrum. Conservative delays correspond to a unitary Koopman operator and preserve the logarithmic scaling due to their equivalence to systems with constant delay, whereas dissipative delays correspond to a non-unitary Koopman operator and change the scaling behavior of the Lyapunov spectrum. All in all, we present an interesting connection between time-delay systems and circle map dynamics, involving the framework of a Koopman operator.