Instability of anchored spirals in geometric flows

TL;DR

This work analyzes anchored rotating spiral waves in a geometric curve evolution model where the normal velocity is $c(\kappa,\kappa_{ss}) = V + D_2\kappa - D_4\kappa_{ss}$. By focusing on the eikonal limit and employing Fenichel geometric singular perturbation theory, the authors prove the existence of rigidly rotating spirals in large annuli and characterize compatibility conditions at the boundaries. They then investigate stability by deriving a slowly varying eigenvalue problem near the eikonal limit and identify a negative threshold for $D_2$ beyond which an edge of the absolute spectrum crosses the imaginary axis, producing an instability that is initially convective and later absolute near the core. Numerical simulations corroborate the theoretical predictions, revealing convective-to-absolute transitions, boundary-layer effects, and subcritical saddle-node bifurcations for finite $\varepsilon$, with implications for wave-train instabilities in reaction-diffusion systems.

Abstract

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.

Figures (10)

• Figure 1: Setup for geometric curve evolution according to \ref{['e:geomlaw']}: shown are unit tangent with a parameterization pointing away from the core anchoring point, the rightward oriented normal, the curvature via inscribed circle of radius $1/|\kappa|$ (left). Note that $\kappa<0$ in this picture, so that the normal speed is less than $V$ when $D_2>0$, effectively shortening and smoothing the curve. Inner and outer boundaries (right) need to be equipped with boundary conditions such as contact angles. Note that $\kappa>0$ near the outer boundary leading to acceleration in this regime when $D_2>0$. Curves converge toward archimedean spirals rotating rigidly with angular frequency $\omega$.
• Figure 2: Schematic of the early stages of the evolution of a straight anchored curve into an Archimedean spiral; boundary condition here is a contact angle $\vartheta=\pi/3$ (top and bottom). This schematic is corroborated in direct numerical simulations with $V=D_2=1,D_4=0$ (bottom).
• Figure 3: Results of direct simulations of \ref{['eq:2.6']} and \ref{['eqn:neumann-bcs']}, with $D_2=D_4=0$ and varying $V=1,5,10$ (top to bottom) and $R_\mathrm{i}=1,5,10$ (left to right). The measured frequencies confirm the relation $\omega=V/R$.
• Figure 4: The limiting frequency is determined by the speed along the boundary, which is determined by the fact that its projection on the normal direction equals $V$ (left). acute contact angles allow for monotone $\Phi(r)$, whereas obtuse angles induce turning points and high curvature through necessary boundary layers (right).
• Figure 5: Flows of \ref{['eq:3.3']} in $\mathbb{R}^4$ for fixed $\Omega$ with $\varepsilon=0$ (left) and $\varepsilon\gtrsim 0$ (right). The vertical hyperplanes $\alpha=const$ are invariant in the left panel and the quarter circle consists entirely of equilibria with 2d strong unstable and 1d strong stable manifolds. In the right figure, the manifold of equilibria is a center manifold with slow drift towards decreasing $\hat{\alpha}$. Strong stable and unstable manifolds continue as invariant foliations for $\varepsilon\gtrsim 0$. Also shown in magenta are sample trajectories off the slow manifold.The boundary condition is a line inside fixed $\hat{\alpha}=1/\hat{R}_\mathrm{i}$.

Theorems & Definitions (6)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Remark 3.4