Dynamical scalarization in Einstein-Maxwell-dilaton theory

We study the process of fully nonlinear dynamical scalarization starting from a charged black hole or naked singularity in asymptotically flat spacetime Einstein-Maxwell-dilaton theory. Initially dilaton field is negligible compared to gravitational and Maxwell field. Then experiences an immediate growth, later it oscillates with damping amplitude finally settles down finite value. For hairy develops original Reissner-Nordstr\"om hole, since oscillation decay are almost independent coupling parameter, unlike Anti-de Sitter not easy distinguish resulting hole. evolves singularity, has rich structures. In process, soon enveloped by one outer horizon, then another horizon developed end stable forms two horizons degenerate into protect singularity. The mass saturates exponentially scalarization.