Mapping Quasi-Periodic Oscillations to Lyapunov Exponent across Black Hole Thermodynamic Phase Transitions

Abstract

We investigate the thermodynamic phase structure of a nonminimally coupled magnetic AdS black hole through the dynamics of timelike particles. The free energy analysis reveals a Van der Waalslike phase transition characterized by small, intermediate, and large black hole phases. We show that both the Lyapunov exponent of unstable circular orbits and the quasi-periodic oscillation (QPO) frequencies associated with stable circular orbits exhibit clear signatures of underlying thermodynamic phase structure, including first order and critical phase transitions. More importantly, we establish a QPO-Lyapunov exponent mapping and demonstrate that the resulting relation inherits the same thermodynamic branch structure. Although the Lyapunov exponent and QPO frequencies originate from unstable and stable circular orbits, respectively, their correspondence emerges from the common black hole spacetime geometry and remains valid even in the absence of phase transitions. Our results reveal an unexplored connection among orbital instability, QPO phenomenology, and black hole thermodynamics, suggesting a potential observational route for probing chaotic orbital dynamics and thermodynamic phases through QPO measurements.