Using Mathematica(R) to explore the behaviour of phase-locked loops

The motivation for this project was to gain a better understanding of the nonlinear behaviour of a phase-locked loop (PLL) circuit. The existence of chaos in an ordinary PLL circuit being used for frequency demodulation was demonstrated by Endo et al. (1988), and in this paper some of the previous results and recent developments are demonstrated using Mathematica. We have found that investigation of the nonlinear behaviour of the PLL has benefited from inputs from more than one discipline. In this context, an advantage of Mathematica is that it provides a convenient framework in which the results from various disciplines can be exchanged. The physics-based analytical tools can be computed symbolically, the electronics-based behavioural models of circuits can be solved numerically, and the geometric techniques of nonlinear dynamics can be implemented straightforwardly