  The fidelity amplitude is a quantity of paramount importance in echo type
experiments. We use semiclassical theory to study the average fidelity
amplitude for quantum chaotic systems under external perturbation. We explain
analytically two extreme cases: the random dynamics limit --attained
approximately by strongly chaotic systems-- and the random perturbation limit,
which shows a Lyapunov decay. Numerical simulations help us bridge the gap
between both extreme cases.
