  Due to the nature of gravity, non-linear effects are left imprinted in the
quasi-normal modes generated in the ringdown phase of the merger of two black
holes. We offer an analytical treatment of the quasi-normal modes at
second-order in black hole perturbation theory which takes advantage from the
fact that the non-linear sources are peaked around the light ring. As a
byproduct, we describe why the amplitude of the second-order mode relative to
the square of the first-order amplitude depends only weakly on the initial
condition of the problem.
