  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 $\delta \approx 4.669$ in one-dimensional discrete systems.
