Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912568265670656 |
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| author | Straube, Arthur V. Méndez, Guillermo Olicón Winkelmann, Stefanie Höfling, Felix Engel, Maximilian |
| author_facet | Straube, Arthur V. Méndez, Guillermo Olicón Winkelmann, Stefanie Höfling, Felix Engel, Maximilian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07275 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator Straube, Arthur V. Méndez, Guillermo Olicón Winkelmann, Stefanie Höfling, Felix Engel, Maximilian Dynamical Systems Chaotic Dynamics Chemical Physics 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. |
| title | Unfolding the geometric structure and multiple timescales of the urea-urease pH oscillator |
| topic | Dynamical Systems Chaotic Dynamics Chemical Physics |
| url | https://arxiv.org/abs/2508.07275 |