Degenerate Bifurcations and Universal Relaxation Scaling in Black Hole Thermodynamics

We present a dynamical systems approach to black hole thermodynamic criticality based on bifurcation equations. We construct an effective thermodynamic landscape in which black holes relax toward equilibrium fixed points. To describe this process, we introduce a flow parameter $\tau$, interpreted as a phenomenological relaxation time, which governs the approach toward equilibrium configurations in thermodynamic state space. Near critical points, the thermodynamic flow simplifies into universal mathematical forms, which allows different black holes to be grouped into different universality classes based on their critical behaviour. Our analysis further shows critical slowing down, with relaxation timescales determined entirely by the local bifurcation structure.