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arXiv · 2604.01135

Oscillations in a scalar differential equation coupled to a diffusive field

Merlin Pelz·Arnd Scheel
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Abstract

We study the emergence of periodic oscillations through a Hopf bifurcation in a scalar diffusion equation on the half line coupled to a dynamic boundary condition. Our results quantify the effect of delay through the buffering in the diffusive field on boundary kinetics, drawing a parallel to the emergence of oscillations in delay equations. Technically, the Hopf bifurcation occurs in the presence of essential spectrum induced by the diffusive field, preventing a simple approach via center-manifold reduction. The results are motivated by observations in biological systems where dynamic boundary conditions arise when modeling surface dynamics coupled to bulk diffusion.

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Merlin Pelz, Arnd Scheel. 2026-04-01. Oscillations in a scalar differential equation coupled to a diffusive field. https://arxiv.org/abs/2604.01135

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