Scroll filament helical

Fig. 6. The curvature-torsion plane for helical scroll filaments. Helices evolve along semicircles, Equation (33), according to a law (34) for the evolution of torsion with time. Stationary helices exist along the nullcline (dashed curve) given by a(/c, r) = 0. (a) The leading order terms in a(/c, r). Equation (35), predict a parabolic nullcline. Straight, twisted filaments (k = 0, lu = r < 0) are stable if o (0, r) > 0 and unstable if a(0, r) < 0. (b) By including higher order terms in o (/c, r), e.g. Equation (38), we change the shape of the nullcline so that straight filaments with small twist are stable whereas ones with large twist are unstable.

Consider a helical filament supporting a scroll ribbon that is untwisted with respect to the Frenet frame. Let = 7t/2 so that V = B and the scroll ribbon is identical to the Frenet ribbon. In particular, the twist of the scroll ribbon is identical to the torsion of the filament. The ribbon is wrapped flat... 

The study of twisted circular scroll rings is easier than that of helical filaments. For a circular scroll ring of radius a t), curvature is la t), torsion is zero, and the uniform twist rate is locked in since the filament is closed, hence w = (f)s must be an integer multiple of k. The dynamics of such a scroll ring are governed by... 

Compared to scroll rings and helices, the geometry of even the simplest knotted filament (the trefoil) is quite complex. Knotted scroll waves are unknown (or unrecognized) in the BZ reaction, and there have been few thorough studies of knotted scroll wave solutions to reaction-diffusion equations modeling excitable media [26,28]. The analytical theory [32] of invariant knotted solutions to the filament equations (15) is not only difficult but also (probably) inapplicable to the invariant knots that have been computed numerically, because the latter are compact structures whose dynamics seem to be dominated by interactions between the closely spaced segments of the knotted filament. 

---