Piecewise-linear analysis of high-damping chaotic phase-locked loops using Melnikov´s method

Studies the chaotic dynamics observed from a widely used second-order phase-locked loops (PLLs) operating as a frequency modulated demodulator. The authors apply Melnikov´s method to high-damping PLLs to obtain the parameter range where there exist homoclinic points. This implies the PLL is chaotic in such a parameter range in view of the Smale-Birkhoff theorem. In particular, since the current PLL used in practice has a triangular phase detector as its nonlinearity (i.e., a periodic triangular shaped function), one can use piecewise-linear methods to derive the Melnikov integral analytically even for the practical high-damping (hence, high dissipation) case. The advantage of this method compared to the previous numerical integration method is that all integrations are made analytically, and therefore reliable results can be obtained for all parameter values. The authors have obtained many boundary curves for homoclinic tangency for a wide range of damping coefficients. As a result, some inaccuracies in previous numerical integration results have been detected