Constant frequency resonant drift

It has been long known that spatially uniform, periodic time forcing leads to the motion of a rigidly rotating spiral wave. If the frequency of forcing is close to the rotation frequency of the spiral wave, a circular, large radius ``Larmor''-type drift results (motion equations of the rotation center are similar to those of a charged particle in a magnetic field); if the frequency is equal to the rotation frequency a linear, directed resonant drift in a direction determined by the phase of the forcing results, Agladze et al. [1]; Davydov et al. [13]; Biktashev and Holden [7]; Mantel and Barkley [18]. Figure 2 illustrates the boundary interactions of a resonantly drifting spiral. The closer the resonantly drifting spiral wave approaches the boundary the faster the core rotates (see 2(b) and (d)), hence the greater the changes in the phase at which the constant frequency forcing is applied, and the greater the changes in the direction of the drift. Fig. 2a shows that the tip positions at which the forcing is applied move along a hypocycloid near the boundary. In Fig. 2(a) the rotation of the core and the boundary induced drift are opposite to each other while in (c) these coincide. Figures 2(e) and (f) illustrate the effects of internal boundaries, or holes in the medium. In (e) the resonantly drifting spiral is repelled by the hole of a similar size to the core of the wave, and in 2(f) is captured by the smaller hole.

  
Figure 2: Tip trajectory of a spiral wave with drift induced by spatially uniform periodic stimulation at a constant frequency. In (a), (c), (e), and (f) the * marks the position of the wavetip at the time when the perturbation is applied. (a) and (b) Stimulation with amplitude A = 0.6. (c)-(f) A=2.5. (b) and (d) Dependencies of the instantaneous frequency of the spiral's wavetip rotation on time. "Squares" indicate the values of the instantaneous frequency of the spiral, and the * indicate the stimulation frequency; (b) corresponds to (a); (d) corresponds to (c). (e) Repulsion of the drifting spiral wave by an obstacle in the medium, a hole with radius . (f) Capture of the resonantly drifting spiral wave by a small hole of radius .