Quasi-Normal Modes and Nonlinear Electrodynamics in Black Hole Phase Transitions

We investigate the connection between thermodynamic phase transitions and quasi-normal modes (QNMs) in charged black holes with a positive curvature constant, within the framework of $F(R)$-Euler-Heisenberg gravity. Nonlinear electromagnetic fields lead to rich thermodynamic phase structures and significantly affect the QNMs of massless scalar fields. By analyzing the QNMs spectrum, we find that the transition point marking the disappearance of the divergence in the QNMs slope parameter $K$ aligns with the change of the thermodynamic phase structure described by the heat capacity, within the bounds of computational uncertainty. This precise matching holds under variations of the curvature parameter and charge. Furthermore, we show that larger angular quantum number $l$ diminishes this correspondence, while higher overtone number $n$ restores it beyond a threshold. These findings demonstrate that thermodynamic phase transitions of black holes carry embedded dynamical information, uncovering a fundamental link between black hole thermodynamic and dynamical properties.