  The ringdown gravitational wave signal arising e.g., in the final stage of a
black hole binary merger, contains important information about the properties
of the remnant, and can potentially be used to perform clean tests of general
relativity. However, interpreting the ringdown signal, in particular when it is
the loudest, requires understanding the role of nonlinearities and their
potential impact on modelling this phase using quasinormal modes. Here, we
focus on a particular nonlinear effect arising from the change in the black
hole's mass and spin due to the partial absorption of a quasinormal
perturbation. We isolate and systematically study this third-order, secular
effect by evolving the equations governing linear metric perturbations on the
background of a spinning black hole, but allowing the properties of the
background to evolve in a prescribed way. We find that this leads to the
excitation of quasinormal modes with higher polar angular number, retrograde
modes (counter-rotating with respect to the black hole), and overtones, as well
as giving rise to a component of the signal at early times that cannot be fully
described using quasinormal modes. Quantifying these effects, we find that they
may be relevant in analyzing the ringdown in black hole mergers.
