Unfolding the Geometric Structure and Multiple Timescales of the Urea–Urease pH Oscillator

We study a two-variable dynamical system modeling pH oscillations in the urea–urease reaction within giant lipid vesicles — a problem that intrinsically contains multiple, well-separated timescales. Building on an existing, deterministic formulation via ordinary differential equations, we resolve different orders of magnitude within a small parameter and analyze the system’s limit cycle behavior using geometric singular perturbation theory (GSPT). By introducing two different coordinate scalings — each valid in a distinct region of the phase space — we resolve the local dynamics near critical fold points, using the extension of GSPT through such singular points due to Krupa and Szmolyan. This framework enables a geometric decomposition of the periodic orbits into slow and fast segments, which allows us to obtain a closed-form estimate for the oscillation period. Moreover, we link the existence of pH oscillations to an underlying biochemical asymmetry, namely, the differential transport across the vesicle membrane.