Analytic structure of phase-locked loops in complex time

The analytic structure of the governing equation for a 2nd order Phase-Locked Loops (PLL) is studied in the complex time plane. By a local reduction of the PLL equation to the Riccati equation, the PLL equation is analytically shown to have singularities which form a fractal structure in the complex time plane. Such a fractal structure of complex time singularities is known to be characteristic for nonintegrable, especially chaotic systems. On the other hand, a direct numerical detection of the complex time singularities is performed to verify the fractal structure. The numerical results show the reality of complex time singularities and the fractal structure of singularities on a curve.