A STABLE KNOTTED SINGULARITY IN AN EXCITABLE MEDIUM

Author:

HENZE CHRIS1,WINFREE ART1

Affiliation:

1. Dept. Ecology and Evolutionary Biology, University of Arizona, USA

Abstract

Numerical studies of an apparently stable knotted singularity are used here to seek out regularities of vortex dynamics in three-dimensional excitable media. Initial conditions were contrived to produce a vortex filament in the form of a trefoil (3:2 torus) knot, using the piecewise linear 'B-kinetics', with excitability parameter 'g' set to 0.9. The simulation was pursued for 3000 time units, or about 125 wave rotations, during which time the knot was observed to rigidly 'precess' about its symmetry axis, completing one turn every 96 wave rotations, and also translate along that same axis, at about 1 percent of wavespeed. These motions, described with respect to the local Frenet frame, were shown to be highly correlated with local geometry and twist, yielding possible 'laws of filament motion', which might prove viable in other contexts. The trefoil knot is the first organizing center to be studied in detail which exhibits nonuniform twist, and its dynamical behavior supports the notion that the arclength derivative of twist is an important determinant of filament motion, first postulated by Keener in 1988. The knot also exhibited changes in twist distribution, and, to a lesser extent, local geometry, within one wave rotation period, adding an unforeseen complication to efforts at understanding vortex dynamics. The knot is compact, only about a wavelength in radius, so filament interactions may play a prominent role in determining the stability and behavior of the object. The locus of wave collisions was determined for the knot; in places this surface apparently adjoins the filament itself. In the course of this study, two different criteria for locating the filament were compared, and many analytic and graphical utilities were developed, which may prove useful in further studies of three-dimensional excitable media.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modeling and Simulation,Engineering (miscellaneous)

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