Probing Thermodynamic Phase Transitions of 4D R-Charged Black Holes via Lyapunov Exponent

We investigate the thermodynamic phase structure of four-dimensional \textit{R}-charged black holes--characterized by four independent $U(1)$ charges--through the lens of Lyapunov exponents associated with unstable circular orbits of both massless and massive particles. Considering three distinct charge configurations (equal, partially unequal, and fully unequal), we show that the thermal profile of the Lyapunov exponent can effectively probe the black hole phase transitions. Furthermore, we demonstrate that the discontinuous jump in the Lyapunov exponent acts as an order parameter, with the associated critical exponent $\delta = 1/2$ near the critical point--consistent across all charge configurations studied. This supports the proposed universality of the exponent and the broader connection between dynamical chaos and thermodynamic phase transitions.