Chaos from phase-locked loops

The authors present a rigorous analysis and a detailed experimental study of a chaotic attractor observed from a widely-used practical circuit: namely, a second-order phase-locked loop as an FM demodulation. The existence of chaos in this system proven rigorously by using Melnikov´s method, and explicit expressions are derived which specify the possible region of chaos, or, more accurately, the region of existence of the homoclinic orbits. Chaos has been demonstrated in actual experiments in which an IC is used as an FM demodulator. It has also been confirmed by computer simulation. Thus, it has been verified that horseshoe chaos occurs for a wide range of parameter values in practical phase-locked loop FM demodulator systems