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Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator

under review
  • 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 and yields closed-form estimates for the period of oscillation. In particular, we link the existence of such oscillations to an underlying biochemical asymmetry, namely, the differential transport across the vesicle membrane.

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Metadaten
Author:Arthur StraubeORCiD, Guillermo Olicón Méndez, Stefanie Winkelmann, Felix Höfling, Maximilian Engel
Document Type:Article
Year of first publication:2025
ArXiv Id:http://arxiv.org/abs/2508.07275
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