Rotating spirals of curvature flows: a center manifold approach

Abstract

Rigidly rotating spiral wave patterns of Archimedean shape are a prominent phenomenon in the spatiotemporal evolution of planar reaction diffusion systems. A first analysis by Wiener and Rosenblueth [27] was motivated by electrical waves in heart tissue. For experimental evidence of Archimedean spirals in Belousov– Zhabotinsky systems and other excitable chemical reaction media see, for example, [24–26,5,22,20,21]. See also the recent survey [17] and references therein. Fluid convection experiments, although modeled by the Navier–Stokes system rather than by reaction and diffusion, also exhibit dynamically intricate spiral wave patterns; see [19]. For a survey of mathematical issues related to spiral waves in reaction diffusion systems, see [10]. One mathematical approach, initiated already by Wiener, aims at a geometric description of the spatiotemporal dynamics of the sharp wave fronts that seem to emanate from a highly focused “core” region or “tip” and that take the form of an Archimedeam spiral, far away from the tip. This approach, where the wave front is represented simply by a planar curve z = z(s) ∈ R2, parametrized over arclength s ≥ 0, is sometimes called kinematic theory of spiral waves and has been developed on a mostly formal level. See, for example, [15,14,18,17] and references therein. Typically the dynamics of the curve z(s) is then described by a curvature driven flow