Bifurcation diagrams and fractal brain boundaries of phase-locked loop circuits

Several 2-parameter bifurcation diagrams and the boundaries of the basin attraction of various periodic orbits associated with a typical phase-locked loop circuit widely used in modern communication systems are presented. Using these diagrams, the authors have identified and confirmed various routes to chaos reported previously for this circuit. Moreover, they have discovered that over certain regions in the parameter space, the circuit is virtually unpredictable over a very long period of time even though it eventually settles down to a periodic steady state. The long transient is manifested in the form of fractal basin boundaries with self-similar structures repeated at any finer scale of resolution. The results implies that if the phase-locked loop (PLL) circuit is operating within the parameter range which resulted in fractal basin boundaries, it will take an extraordinarily long pull-in time to achieve synchronization. The analysis therefore predicts not only the failure boundaries (when the circuit is chaotic), but also a safety margin for design, namely, that one should stay away from fractal basin boundaries