Anchoring-condition dependence of rotating spiral waves in driven curvature flow: frequency selection for non-orthogonal contact angles and bifurcations from dynamic anchoring

Rigidly rotating spiral waves anchored to a circular core were recently constructed and proved orbitally stable for the driven curvature flow with normal velocity $c=V+D\kappa$ under perpendicular anchoring (contact angle $\pi/2$), together with a large-core expansion of the selected rotation frequency whose leading correction is governed by the first zero $\sigma_0$ of the derivative of the Airy function (Li and Scheel, arXiv:2312.07809). The source leaves open how the anchoring condition controls selection and stability beyond this case: (i) a classification of existence and frequency selection for fixed non-orthogonal contact angles, including the predicted qualitative change of the core boundary layer (the Airy constant $\sigma_0$ replaced by $\sigma_1$, the first zero of the Airy function itself, on one side of the perpendicular case, and a smooth frequency expansion on the other) and possible non-monotone behavior; and (ii) whether dynamic boundary conditions, in which the anchoring contact point or angle evolves, preserve well-posedness and can produce saddle-node or Hopf bifurcations mimicking retracting-wave or meander instabilities of reaction-diffusion spirals. A 2026 follow-up by an overlapping author group proved existence near the eikonal limit (the regime where curvature corrections to the normal velocity are negligible) for contact angles away from the perpendicular case in a fourth-order regularized model, and formally predicted and numerically found Hopf and saddle-node bifurcations driven by negative line tension, but with static boundary conditions, and with the perpendicular case and boundary layers left open. The anchoring-condition problem as stated therefore remains open.

Background

Spiral waves and their kinematic description. Spiral waves are rotating wave patterns that arise in excitable and oscillatory media, for example the Belousov-Zhabotinsky chemical reaction and heart muscle, where they are associated with certain cardiac arrhythmias. In the kinematic theory of such waves, developed quantitatively by Mikhailov and Zykov, the wave front is idealized as a sharp planar curve whose normal velocity is determined by its geometry. The source paper studies the simplest such law, the driven curvature flow

$$c = V + D\kappa,$$

where $V>0$ is the constant normal speed of a straight front, $D>0$ is a line-tension coefficient that penalizes curvature in analogy with surface tension, and $\kappa$ is the signed curvature of the curve.

Anchoring and frequency selection. A spiral tip is typically pinned to a small inhomogeneity or hole in the medium. This is modeled by attaching the curve to the boundary of an excluded core disk of radius $R$ at a prescribed contact angle, the angle at which the curve meets the circle. Rigidly rotating solutions converge at large radius to Archimedean spirals, curves whose successive turns have asymptotically constant radial spacing, rotating with angular frequency $\omega$; a central question of the kinematic theory is which frequency the anchoring condition selects. Li and Scheel (arXiv:2312.07809) settled the perpendicular case (contact angle $\pi/2$, the curve leaving the core circle radially): writing the curve as a graph in polar coordinates and passing to a rotating frame, they reduced existence to a shooting problem for a planar ODE in variables $(\ell,\alpha)$ (tuning initial data so that the orbit satisfies a prescribed far-field condition), with $\alpha=1/r$ and $\ell$ encoding the local wavenumber, the contact angle entering through the boundary value $\ell=\alpha_*\tan\beta$ with $\alpha_*=1/R$ and $\beta$ the deviation from perpendicular. They proved existence and uniqueness of an outward-rotating, asymptotically Archimedean spiral; orbital stability, meaning that perturbed curves stay close to the spiral for all times modulo the rotation, via a comparison principle for curvature flows; and the large-core expansion

$$\omega_{\mathrm{sp}} = VR^{-1} - \sigma_0\,2^{1/3}D^{2/3}V^{1/3}R^{-5/3} + \mathrm{O}(R^{-7/3}),$$

where $\sigma_0=1.01879297\ldots$ is fixed by the first zero of the derivative of the Airy function, $\mathrm{Ai}'(-\sigma_0)=0$. The $R^{-5/3}$ curvature-induced correction was new.

Instabilities of spirals. In full reaction-diffusion systems, rotating spirals can lose stability through meander, in which the tip traces epicycloid-like trajectories, or through drift; Barkley connected these bifurcations to the Euclidean symmetry of the plane, and Wulff, together with Sandstede and Scheel, analyzed them rigorously by center-manifold reduction, a dynamical-systems technique that reduces the dynamics near a non-hyperbolic state to dynamics on a low-dimensional invariant manifold. Those reductions require spectral gaps that fail for Archimedean spirals: the linearization has essential spectrum, the part of the operator spectrum coming from the far field rather than from localized modes, on the imaginary axis, so localized perturbations are transported outward without decaying and the interface does not heal. The source's Discussion therefore proposes two anchoring-related extensions. First, extend the selection analysis to fixed non-orthogonal contact angles: the paper asserts that the existence argument generalizes readily with boundary condition $\ell=\alpha_*\tan\beta$ and describes, without proof, how the asymptotics change, with the fully developed core boundary layer replacing $\sigma_0$ by $\sigma_1$ (the first zero of the Airy function itself) on one side of $\beta=0$ and a smooth expansion $\omega_*=V\alpha_*+\mathrm{O}(\alpha_*^2)$ on the other, the boundary layer being induced by a non-monotonicity of the angle profile in the radius. Second, study more general, possibly dynamic, boundary conditions, in which the attachment point or angle evolves according to additional physical processes, asking whether they preserve well-posedness and can generate saddle-node bifurcations (two steady states colliding and disappearing) or Hopf bifurcations (a complex-conjugate pair of eigenvalues crossing the imaginary axis and creating oscillations), thereby mimicking the retracting-wave and meandering instabilities of reaction-diffusion spirals (a retracting wave being an anchored front that recedes along its anchor instead of emitting outward; meandering, the tip tracing compound epicycloid-like paths).

Follow-up work. Cortez, Li, Mihm, Xu, Yu and Scheel (Physica D 490 (2026) 135159) extended the model to $c=V+D_2\kappa-D_4\kappa_{ss}$, adding a fourth-order regularization with $D_4>0$ ($\kappa_{ss}$ denoting the second arclength derivative of the curvature) and allowing negative line tension $D_2<0$, which destabilizes the emitted wave trains against transverse modulations. Near the eikonal limit (the regime of dominant constant normal speed, where curvature corrections are small) and for an open class of compatible static boundary conditions they proved existence of rotating spirals with inner contact angle $\vartheta_{\mathrm{i}}\in(0,\pi/2)$ and frequency $\omega=V\sin(\vartheta_{\mathrm{i}})^{-1}R_{\mathrm{i}}^{-1}+\mathrm{O}(R_{\mathrm{i}}^{-2})$, formally predicted a Hopf bifurcation at a critical negative $D_2$ with eigenfunctions growing super-exponentially toward infinity, and found convective and absolute oscillatory instabilities and saddle-node bifurcations numerically. Notably, the perpendicular angle $\vartheta_{\mathrm{i}}=\pi/2$ is a singular excluded case in their analysis, and boundary layers, dynamic boundary conditions, non-graph curves, and diffusive inter-arm coupling all remain open.

Problem Statement

For the driven curvature flow of a planar curve with normal velocity $c=V+D\kappa$ ($V>0$ a constant driving speed, $D>0$ a line-tension coefficient, $\kappa$ the signed curvature), attached to the boundary circle $r=R$ of a core region, the source established existence, uniqueness, orbital stability, and the large-core frequency expansion only for perpendicular anchoring (contact angle $\pi/2$, deviation $\beta=0$). Resolve how the anchoring condition controls selection and stability beyond this case.

(1) Static anchoring classification. For fixed non-orthogonal contact angles (the source's boundary condition $\ell=\alpha_*\tan\beta$ with $\alpha_*=1/R$), classify existence, uniqueness, and multiplicity of rigidly rotating (turning at constant angular speed without change of shape), asymptotically Archimedean (successive turns approaching constant radial spacing at large radius) spiral waves as a function of $\beta$, and determine the selected rotation frequency $\omega_*(\beta)$ in the large-core limit $R\to\infty$: for which signs and ranges of $\beta$ the core boundary layer is fully developed with the constant $\sigma_1$ (first zero of the Airy function) replacing $\sigma_0$ (first zero of its derivative), for which the frequency admits a smooth expansion, and how the non-monotonicity of the angle profile induces the boundary layer.

(2) Dynamic anchoring. Formulate anchoring conditions in which the attachment point or contact angle evolves according to an additional physical law rather than being fixed, preserving well-posedness of the curvature-flow PDE, and determine whether such dynamic anchoring conditions give rise to saddle-node bifurcations (two steady states colliding and disappearing) or Hopf bifurcations (a complex-conjugate pair of eigenvalues crossing the imaginary axis, creating oscillations) of the rotating spiral waves, thereby mimicking retracting-wave or meandering (tip tracing compound, epicycloid-like paths rather than a circle) instabilities of spirals in reaction-diffusion systems.

The common objective is a characterization of how the anchoring condition at the core boundary determines selection and stability of rotating spiral waves in the geometric model: component (1) fixes the static baseline of frequency selection as a function of anchoring geometry, and component (2) asks the dynamical question whether anchor dynamics alone can nucleate the instabilities observed in full reaction-diffusion systems. Note that Hopf or saddle-node bifurcations produced by bulk negative line tension with static boundary conditions, found in the 2026 follow-up, do not by themselves answer component (2).

Component (2) concerns a dynamic anchoring law, governing the attachment point or contact angle, that is faithful to the geometric model — its static limit is a fixed contact angle, and it is formulated without reference to the bifurcation question it is meant to answer — and for which the curvature-flow problem is well-posed; it asks whether such a law produces a saddle-node bifurcation, a Hopf bifurcation, or neither.

References

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2. Anthony Cortez, Nan Li, Nathan Mihm, Alice Xu, Xiaoxing Yu, and Arnd Scheel, 'Instability of anchored spirals in geometric flows', Physica D: Nonlinear Phenomena 490 (2026) 135159, DOI 10.1016/j.physd.2026.135159 (preprint arXiv:2504.07270), https://arxiv.org/abs/2504.07270
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