On the Stability of Knots in Excitable Media

Abstract

Through extensive numerical simulations we investigate the evolution of knotted and linked vortices in the FitzHugh-Nagumo model. On medium time scales, of the order of a hundred times the vortex rotation period, knots simultaneously translate and precess with very little change of shape. However, on long time scales we find that knots evolve in a more complicated manner, with particular arcs expanding and contracting, producing substantial variations in the total length. The topology of a knot is preserved during the evolution and after several thousand vortex rotation periods the knot appears to approach an asymptotic state. Furthermore, this asymptotic state is dependent upon the initial conditions and suggests that, even within a given topology, a host of meta-stable configurations exist, rather than a unique stable solution. We discuss a possible mechanism for the observed evolution, associated with the impact of higher frequency wavefronts emanating from parts of the knot which are more twisted than the expanding arcs.