Critical phenomena in dynamical scalarization with a massive and self-interacting scalar field

We conduct numerical simulations on the scalarization of charged black holes, uncovering the crucial role of the nonlinear terms ($\ensuremath{\lambda}{\ensuremath{\phi}}^{4})/2+({\ensuremath{\mu}}^{2}{\ensuremath{\phi}}^{2})/2$ in the critical phenomena of black hole dynamical scalarization. Our findings indicate that the critical parameters exhibit a linear dependence on $\ensuremath{\lambda}$ and ${\ensuremath{\mu}}^{2}$. We particularly emphasize the temporal patterns of the critical solution and the power-law behavior of black hole mass in critical phenomena. Importantly, we observe that in the relationship between the time spent on the critical solution and the initial parameters, both subcritical and supercritical cases follow the same pattern, differing only in the constant within the linear relationship. In all cases, the black hole mass shows a power-law behavior, although with different exponents.