Quantized Spiral Tip Motion in Excitable Systems with Periodic Heterogeneities

Abstract

Meandering spiral tips in homogeneous reaction-diffusion systems are characterized by two generically incommensurate radii and frequencies. Here, we create periodic perturbations in space to induce a transition to commensurate radii and frequencies that exhibit a devil's staircase. The plateaus of the staircase correspond to pinned or complex periodic orbits of the spiral tip.

Pinning of vortex rings and vortex networks in excitable systems

“…Over the past two decades systematic research efforts have resulted in a very good understanding of two-dimensional spirals and their response to external fields and gradients [6,7]. Also the pinning of these vortices to unexcitable domains has attracted considerable interest [8][9][10] which is partly driven by its relevance to cardiology and specifically tachycardia and sudden cardiac death [11,12]. The first, controlled experimental examples of spiral wave pinning were reported for the photosensitive, Ru(bpy) 3 -catalyzed Belousov-Zhabotinsky (BZ) reaction, where an argon ion laser was used to created photochemically inhibited disks [13].…”

“…For smaller orbits, (normal) dispersion decreases the wave speed and, hence, causes periods larger than T 0 [13]. Chemical wave rotation around even smaller obstacles (U ≪ λ r ) were studied in microfluidic devices with hundreds of inert (a) E-mail: steinbck@chem.fsu.edu obstacles arranged on a square lattice [8]. These conditions induce rotation around obstacle groups in which the spiral tip traces complex periodic orbits and the U -dependence of the period follows a devil's staircase.…”

Pinning of scroll waves to flat and highly branched unexcitable heterogeneities

“…This case should be particularly interesting if some of the holes are sufficiently small to block wave passage according to firing sequences such as passage followed by blockage (1:1) or more complex sequences [28,29]. Other more obvious variations should explore different arrangements of the holes including square patterns and nested vortices.…”

Introduction: Self-organization in nonequilibrium chemical systems

“…62 For the BZ reaction, Ginn et al explored the use of soft lithography to control wave propagation through channel networks as narrow as 50 m. 63 They also investigated the dynamics of spiral wave rotation in grid-like networks of excitable channels and reported sequences of complex tip trajectories. 64 Pattern-forming systems with even smaller reactor units have been developed and studied at Brandeis University. Vanag, Epstein and co-workers have investigated the BZ reaction in microemulsions consisting of small aqueous BZ droplets surrounded by a monolayer of the surfactant AOT in a continuous oil phase.…”