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We study the process of fully nonlinear dynamical scalarization starting froma charged black hole or a naked singularity in asymptotically flat spacetime inthe Einstein-Maxwell-dilaton theory. Initially the dilaton field is negligiblecompared to the gravitational and the Maxwell field. Then the dilaton fieldexperiences an immediate growth, later it oscillates with damping amplitude andfinally settles down to a finite value. For a hairy black hole develops from anoriginal Reissner-Nordstr\"om black hole, since the dilaton oscillation anddecay are almost independent of the coupling parameter, unlike the Anti-deSitter spacetime it is not easy to distinguish the resulting hairy black holefrom the original asymptotically flat charged hole. For a hairy black holeevolves from an original naked singularity, the resulting hairy black hole hasrich structures. In the scalarization process, the naked singularity is soonenveloped by one outer horizon, then another horizon is developed and in theend a stable hairy black hole forms and two horizons degenerate into one toprotect the singularity. The hairy black hole mass saturates exponentially inthe scalarization.

Dynamical scalarization in Einstein-Maxwell-dilaton theory