Spiral Wave Meander and Symmetry of the Plane

Abstract

We present a general group-theoretic approach that explains the main qualitative features of the meander of spiral wave solutions on the plane. The approach is based on the well-known space reduction method and is used to separate the motions in the system into superposition of those ‘along’ orbits of the Euclidean symmetry group, and ‘across’ the group orbits. It can be interpreted as passing to a reference frame attached to the spiral wave’s tip. The system of ODEs governing the tip movement is obtained. It is the system that describes the movements along the group orbits. The motions across the group orbits are described by a PDE which lacks the Euclidean symmetry. Consequences of the Euclidean symmetry on the spiral wave dynamics are discussed. In particular, we explicitly derive the model system for bifurcation from rigid to biperiodic rotation, suggested earlier by Barkley [1994] from a priori symmetry considerations.

Freezing Solutions of Equivariant Evolution Equations

“…Freezing such waves can be delicate because it depends on the precise choice of phase condition (with or without weighted L 2 -norms), the type of numerical discretization (rectangular or polar grid), and on the right choice of the underlying group. Note that a related approach to ours was developed in [5] with the intention of using the side constraint in order to fix the tip of a spiral wave. Also in [2] Barkley mentions the use of a pinning condition or phase condition in order to compute the spiral wave from a time-independent boundary value problem.…”

Effective dynamics of twisted and curved scroll waves using virtual filaments

“…This regime is called meandering and has been related to the symmetries of the Euclidean plane in Refs. [29,[55][56][57]. In some detailed models of cardiac excitation, the diffusive properties around the spiral tip are even more pronounced, such that eventually only the refractoriness of previously excited tissue governs the tip trajectory; this regime delivers so-called linear cores [49].…”

Spontaneous Halt of Spiral Wave Drift in Homogeneous Excitable Media

“…Ротор представляет собой вращающуюся фазовую волну химической или любой другой активности, которая распространяется по стационарной среде. В однородной среде ротор обычно имеет вид спирали Архимеда, вращающейся с постоянной скоростью [1].При определенных упрощениях спиральную волну часто бывает полезно представить как искривленную полуволну. Обрыв этой полуволны носит название кончика спиральной волны.…”

“…Ротор представляет собой вращающуюся фазовую волну химической или любой другой активности, которая распространяется по стационарной среде. В однородной среде ротор обычно имеет вид спирали Архимеда, вращающейся с постоянной скоростью [1].…”

“…Согласно исследованиям некоторых авторов, смена типа поведения кончика ротора с равномерного кругового вращения на меандр и с меандра на гипермеандр обусловлены бифуркацией Андронова-Хопфа [1].…”