Winfree Meandering in a 2-Dimensional 2-Variable Excitable Medium

Abstract

Two-dimensional excitable media, for example the excitable version of the Belousov-Zhabotinsky reaction, are capable of forming spiral-shaped self-sustaining rotating wave patterns (rotors). In order to explain Winfree's experimental observation of an irregular "meandering" of the rotor's core region, we present a numerical simulation of a continuous, two-variable excitable medium in two space dimensions. Two phenomena occur: 1) an irregular motion of the rotor's core; 2) a non-stationary peak inside the core region. Thus, "meandering" is obtained, together with a new phenomenon, the "peak". A sufficient condition for both phenomena is that the underlying (local) system be stiff, that is, admit two time scales for its approximate description. In this case, (1) the "reaching distance" of diffusion is small as compared with the core's radius, and (2) rotation symmetry of the core implies a gradation in local frequencies (increase toward the center), supposing radial decoupling. We propose that both constraints act together to induce a spontaneous breakdown of rotation symmetry under an increase of stiffness. Immediately (or soon) thereafter, the core is no longer synchronized by the wave circling around it; instead, excitation from the surrounding wave penetrates the core along a new path from time to time, causing a non-repetitive short-cut.

A Cellular Automaton Model of Excitable Media Including Curvature and Dispersion

“…4 rotates more or less rigidly around a pivot point, but for other parameter values we have observed (i) meandering spiral cores (19) and (ii) chaotic self-reproduction of spirals by spontaneous wave-breaking (1,20) (Fig. 5).…”

Chemical Oscillations, Waves, and Turbulence

“…6.6. on the basis of the Ginzburg-Landau equation, but we were unable to analyze their stability because of the mathematical difficulties involved. Moreover, there exists the opinion that the core meandering is a form of diffusion-induced chemical turbulence (Rössler and Kahlert, 1979). It is reported (Winfree, 1978), however, that more careful observations reveal that the center of the core is not stricdy fIXed but meanders in a rather irregular way.…”

Wave-particle dualism of spiral waves dynamics

“…An ODE description is used in the theory of meander, i.e. non-stationary rotation of spiral waves, which is possible in some reaction-diffusion systems even in the absence of any perturbations [16][17][18]. The description of these complex motions turned from pure phenomenology to theory after the discovery that the transition from stationary rotation to biperiodic meander happens as if it was a Hopf bifurcation [19,20].…”