Anchored spirals in the driven curvature flow approximation

Abstract

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. \end{abstract}

Figures (5)

• Figure 1: Schematic picture of spiral filament anchored at a disc of radius $R$, evolving pointwise with velocity $\gamma_t$, with effective normal velocity given by $V+\kappa D$. The normal motion can lead to an effective rigid rotation with angular frequency $\omega$.
• Figure 2: (a) Invariant regions $A_\omega$ and $B_\omega$ on the phase plane. The trajectory of $\Psi_{\tau}(\alpha_*,0;\omega)$ stays in the region $G_\omega$ for $\tau > 0$. (b) The $\ell$-coordinate of equilibrium point $\omega_*$ is strictly increasing in $\alpha_*$; the region below the trajectory at $\omega_*$ is forward invariant for $\hat{\omega}>\omega_*$ (indicated by blue arrows) showing that the associated $\hat{\alpha}$ necessarily satisfies $\hat{\alpha}>\alpha_*$; similarly the region above the trajectory at $\omega_*$ is forward invariant for $\tilde{\omega}<\omega_*$ (indicated by red arrows) showing that the associated $\tilde{\alpha}$ necessarily satisfies $\tilde{\alpha}<\alpha_*$.
• Figure 3: Figure (a): The phase portrait of (\ref{['vf omega']}) with $\omega=0, V=D=1$ and its invariant manifold $\mathcal{M}_0$. The point $(\ell_1,\alpha_1) = (0,1)$ is not normally hyperbolic. Figure (b): The invariant manifold $\mathcal{M}_\omega$ where $\omega > 0$.
• Figure 4: Graphs of spirals with various core radii in the plane (left) and in $r-\phi$-plots; for all graphs, $V = D = 1$; (a)-(b): $R = 5$, (c)-(d): $R=50$.
• Figure 5: Comparison of the theoretical prediction from Theorem \ref{['theorem existence in polar coordidate']} and computational results with $V=D=1$. Figure (a): Theoretical expansion $\omega=VR^{-1}-\sigma_0 2^{1/3} D^{2/3} V^{1/3} R^{-5/3}$ (blue solid line) and the numerical results (red dotted line). This shows $\omega = R^{-1}$ asymptotically. Figure (b): Theoretical expansion $\omega-VR^{-1}=-\sigma_0 2^{1/3} D^{2/3} V^{1/3} R^{-5/3}$ (blue solid line) and the numerical results (red dotted line). The $\log$-$\log$ plots coincide on a line of slope $-5/3$ which validates the expansion $\omega = V R^{-1} + \mathcal{O}(R^{-5/3})$. We confirmed that the difference between numerical results and the theoretical prediction in the right-hand plot is of order $R^{-7/3}$.

Theorems & Definitions (11)

• Theorem 1.1: Existence
• Theorem 1.2: Asymptotics --- large core
• Theorem 1.3: Stability --- Informal
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• Theorem 4.1
• proof : Proof of Theorem \ref{['theorem existence in polar coordidate']}