  The Costas loop is a modification of the phase-locked loop circuit, which
demodulates data and recovers carrier from the input signal. The Costas loop is
essentially a nonlinear control system and its nonlinear analysis is a
challenging task. Thus, simplified mathematical models and their numerical
simulation are widely used for its analysis. At the same time for phase-locked
loop circuits there are known various examples where the results of such
simplified analysis are differ substantially from the real behavior of the
circuit. In this survey the corresponding problems are demonstrated and
discussed for the QPSK Costas loop.
