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Instability of anchored spirals in geometric flows
We investigate existence, stability, and instability of anchored rotating spiral waves in a model for geometric curve evolution. We find existence in a parameter regime limiting on a purely eikonal curve evolution. We study stability and instability both theoretically in this limiting regime and numerically, finding both oscillatory, at first convective instability, and saddle-node bifurcations. Our results in particular shed light onto instability of spiral waves in reaction-diffusion systems caused by an instability of wave trains against transverse modulations.
@article{cortez2025instabilityanchoredspiralsgeometric,
title={Instability of anchored spirals in geometric flows},
author={Anthony Cortez and Nan Li and Nathan Mihm and Alice Xu and Xiaoxing Yu and Arnd Scheel},
year={2025},
eprint={2504.07270},
archivePrefix={arXiv},
primaryClass={nlin.PS},
url={https://arxiv.org/abs/2504.07270},
} 
Anchored spirals in the driven curvature flow approximation
We study existence, asymptotics, and stability of spiral waves in a driven curvature approximation, supplemented with an anchoring condition on a circle of finite radius. We analyze the motion of curves written as graphs in polar coordinates, finding spiral waves as rigidly rotating shapes. The existence analysis reduces to a planar ODE and asymptotics are given through center manifold expansions. In the limit of a large core, we find rotation frequencies and corrections starting form a problem without curvature corrections. Finally, we demonstrate orbital stability of spiral waves by exploiting a comparison principle inherent to curvature driven flow.
@article{li2024,
title={Anchored spirals in the driven curvature flow approximation},
author={Nan Li and Arnd Scheel},
year={2024},
eprint={2312.07809},
archivePrefix={arXiv},
primaryClass={nlin.PS},
url={https://arxiv.org/abs/2312.07809},
}