title: Analytical and numerical investigations of the phase-locked loop with time delay.
abstract: We derive the normal form for the delay-induced Hopf bifurcation in the first-order phase-locked loop with time delay by the multiple scaling method. The resulting periodic orbit is confirmed by numerical simulations. Further detailed numerical investigations demonstrate exemplarily that this system reveals a rich dynamical behavior. With phase portraits, Fourier analysis, and Lyapunov spectra it is possible to analyze the scaling properties of the control parameter in the period-doubling scenario, both qualitatively and quantitatively. Within the numerical accuracy there is evidence that the scaling constant of the time-delayed phase-locked loop coincides with the Feigenbaum constant delta approximately 4.669 in one-dimensional discrete systems.
authors.affiliation: Institute of Parallel and Distributed Systems (IPVS), University of Stuttgart, Breitwiesenstrasse 20-22, D-70565 Stuttgart, Germany. michael.schanz@informatik.uni-stuttgart.de
authors.forename: Michael
authors.initials: M
authors.lastname: Schanz
authors.forename: Axel
authors.initials: A
authors.lastname: Pelster