# Oscillations in a scalar differential equation coupled to a diffusive field

Source: [Oscillations in a scalar differential equation coupled to a diffusive field](<https://www.sota2.com/research/paper/69cdf8240636e5bf4afccbd8>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2026-04-03T05:00:12.986Z

Authors: Merlin Pelz, Arnd Scheel

Published: 2026-04-01T16:53:46.000Z

## 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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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2604.01135>)

[PDF](<https://arxiv.org/pdf/2604.01135.pdf>)