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Analytical and Numerical Investigation of the Phase-Locked Loop with Time Delay

Michael Schanz·Axel Pelster
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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 $δ\approx 4.669$ in one-dimensional discrete systems.

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Michael Schanz, Axel Pelster. 2005-01-14. Analytical and Numerical Investigation of the Phase-Locked Loop with Time Delay. https://doi.org/10.1103/physreve.67.056205

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