Analytical and numerical investigations of the phase-locked loop with time delay

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.