In the present paper we show the existence of a fully nonlinear dynamical mechanism for the formation of scalarized black holes which is different from the spontaneous scalarization. We consider a class of scalar-Gauss-Bonnet gravity theories within which no tachyonic instability can occur. Although the Schwarzschild black holes are linearly stable against scalar perturbations, we show dynamically that for certain choices of the coupling function they are unstable against nonlinear scalar perturbations. This nonlinear instability leads to the formation of new black holes with scalar hair. The fully nonlinear and self-consistent study of the equilibrium black holes reveals that the spectrum of solutions is more complicated possessing additional branches with scalar field that turn out to be unstable, though. The formation of scalar hair of the Schwarzschild black hole will always happen with a jump because the stable scalarized branch is not continuously connected to Schwarzschild one.
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