  Within linearized perturbation theory, black holes decay to their final
stationary state through the well-known spectrum of quasinormal modes. Here we
numerically study whether nonlinearities change this picture. For that purpose
we study the ringdown frequencies of gauge-invariant second-order gravitational
perturbations induced by self-coupling of linearized perturbations of
Schwarzschild black holes. We do so through high-accuracy simulations in the
time domain of first and second-order Regge-Wheeler-Zerilli type equations, for
a variety of initial data sets. We consider first-order even-parity
$(\ell=2,m=\pm 2)$ perturbations and odd-parity $(\ell=2,m=0)$ ones, and all
the multipoles that they generate through self-coupling. For all of them and
all the initial data sets considered we find that ---in contrast to previous
predictions in the literature--- the numerical decay frequencies of
second-order perturbations are the same ones of linearized theory, and we
explain the observed behavior. This would indicate, in particular, that when
modeling or searching for ringdown gravitational waves, appropriately including
the standard quasinormal modes already takes into account nonlinear effects.
