Any simulator will solve a number of ordinary diferential equations (ODEs) given by Kirckoff laws over all nodes. Those equations can be writen in time (Transient analysys) or in frequency (AC analysys)
Because solutions for both .TRAN and .AC depend on initial conditions (IC), those IC will be found first, to have an initial point to start solving the equations(DC analysys).
Because frequency equations are very complicated (those H(j*Omega)), a linearization of the Time ODEs around IC is performed first.
Then a "small" sinusoidal signal (whith variable frequency) is aplied at entry to observe the output.
This is how .AC analysys is performed
Because a linearisation was performed first, there is no diference between input and output frequency in .AC analysys