BIFURCATION DIAGRAMS AND FRACTAL BASIN BOUNDARIES OF PHASE-LOCKED LOOP CIRCUITS

We present several 2-parameter bifurcation diagrams and the boundaries of the basin of attraction of various periodic orbits associated with a typical phase-locked loop circuit widely used in modern communication systems. Using these diagrams, we have identified and confirmed various different routes to chaos reported previously for this circuit. Moreover, we 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. This result 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 extraordinaril? long pull-in time to achieve synchronization. Our analFsis predicts therefore not only the failure boundaries (when the circuit is chaotic), but also a safety margin for design; namely, stay away from fractal basin boundaries. © 1990 IEEE