Dynamics of spontaneous scalarization of black holes with nonlinear electromagnetic fields in anti-de Sitter spacetime

We investigate spontaneous scalarization in the Einstein-Born-Infeld-Scalar (EBIS) model with asymptotically AdS boundary conditions, revealing novel dynamical critical phenomena in black hole evolution. Through numerical analysis, we discover a distinctive ``flip" phenomenon where the scalar field exhibits critical transitions between different stable configurations. These transitions manifest in two forms: a single flip under variations in initial perturbation amplitude or scalar-electromagnetic coupling, and a double flip when varying black hole charge. Near critical points, the system displays universal relaxation behavior characterized by logarithmic scaling of relaxation time, $\tau \propto \ln |p - p_s|$, where $p_s$ denotes the critical initial amplitude. We demonstrate that these transitions arise from the system's approach to unstable AdS-Born-Infeld black hole configurations, which serve as separatrices between distinct stable phases. The Born-Infeld parameter plays a crucial role in this dynamics, with scalar hair vanishing in the strong nonlinearity limit. These results reveal fundamental aspects of black hole phase transitions in theories with nonlinear electromagnetic couplings and provide new insights into critical phenomena in gravitational systems.