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 r eduction of the PLL equation to the Ricatti equation, the PLL equation is analytically shown to have singularities which form a fractal stru cture in the complex time plane. Such a fractal structure of complex t ime singularities is known to be characteristic for nonintegrable, esp ecially chaotic systems. On the other hand, a direct numerical detecti on of the complex time singularities is performed to verify the fracta l structure. The numerical results show the reality of complex time si ngularities and the fractal structure of singularities on a curve.