Drift velocity field

It is important to stress that under the described one-channel feedback the drift direction does not depend on the initial orientation o of the spiral. The location 0 of the spiral center and the time delay r completely determine the drift angle 7. Thus, the drift velocity field induced by one-channel feedback control follows from Eq. (9.14) ... 

This drift velocity field is shown in Fig. 9.2 corresponding to r = 0. The constant ip is taken as (/> == —1.8 in accordance with the chosen parameters for the Oregonator model (9.1). The field has rotational symmetry, but the drift angle 7 monotonously increases with the distance z from the detector point. Hence, there is a discrete set of sites along any radial direction, where the drift direction is orthogonal to the radial one i.e. 

Fig. 9.2. Drift velocity field under one-channel feedback for t = 0. The thick solid line shows the trajectory of the spiral center computed from the Oregonator model (9.1). ...

Fig. 9.4(b) shows the spiral tip trajectory obtained experimentally under this feedback control. After a short transient the spiral core center drifts in parallel to the line detector. The asymptotic drift trajectory reminds the resonance attractor observed under one-channel control, because a small variation of the initial location of the spiral wave does not change the final distance between the detector and the drift line. To construct the drift velocity field for this control algorithm an Archimedean spiral approximation is used again. Assume the detector line is given as a = 0 and an Archimedean spiral described by Eq. (9.5) is located at a site x,y) with a > 0. A pure geometrical consideration shows that the spiral front touches the detector each time ti satisfying the following equation ... 

The obtained drift velocity field is shown in Fig. 9.5(a). In analogy to one-channel feedback, there is a set of stationary trajectories given here by the condition... 

An arbitrary curved line detector should be considered as a set of continuously connected arcs with different curvature radii. The determination of the drift velocity field in this case is illustrated in Fig. 9.9 for a detector... 

To confirm these conclusions we have performed numerical simulations of the feedback mediated drift within the Oregonator model (9.1). The thick solid lines in Fig. 9.9 show the obtained trajectories of a spiral wave center for two different initial locations. The spiral center moves in very good agreement with the predicted drift velocity field and stops near the place where the velocity field vanishes. 

Fig. 9.9. Drift velocity field for a feedback loop based on a strongly curved detector (dashed line) and corresponding trajectories of the spiral center computed for the Oreg-onator model (9.1). r = 1.4.

The constant (/ specifies the drift direction induced in the case r = 0 and 0 = 0. In the Oregonator model (9.1) one finds with the parameters indicated before ip = —0.5 [47]. Hence, the drift velocity field can be written as ... 

To determine the drift velocity field for an arbitrarily shaped spatial domain, we divide it into a set of small subdomains. Each subdomain is treated as a one-point detector generating a feedback signal with a phase shift given by Eq. (9.46). Finally, the resulting drift velocity field is derived according to Eqs. (9.45) and (9.42) as the sum over all drift vectors induced by single subdomains. This superposition principle unifies and simplifies the study of different algorithms for continuous feedback control considerably. 

The amplitude A z) and the phase 4> z) of the drift velocity field induced by the two points together are determined from Eq. (9.45) as... 

To analyze the obtained drift velocity field it is suitable to choose the distance between the two measuring points dp = 2a as control parameter [53]. For dp/X 1, the drift velocity field looks very similar to that induced by one-point feedback (compare section 9.3.1, [47]). It includes a set of circular-shaped attracting manifolds called resonance attractors [21], as shown in Fig. 9.2. This attractor structure still persists for distances dp/X < 0.5. For example, the drift velocity field obtained for dp/X = 0.45 is shown in Fig. 9.10(a). The thick solid line represents the drift... 

Fig. 9.10. Drift velocity field determined from Eqs.(9.43), (9.42), (9.48) for (a) dp/A = 0.45, (b) dp/X = 1.0. Thin solid lines represent lines of fixed points that satisfy Eq. (9.49) (compare text). Thick solid lines depict trajectories of the spiral center computed for the Oregonator model (9.1) with fc/ , = 0.02 and t = 0 [53].

For 0.5 < dp/X < 1.5, the drift velocity field changes dramatically, see Fig. 9.10(b). There are three spatially unbounded fixed lines that destroy the circular-shaped attractors existing as long as dp/ < 0.5. In numerical simulations with the Oregonator model (9.1), the spiral wave center follows an approximately circular trajectory until it stops practically at a fixed line in complete agreement with the predictions from the drift velocity field. 

Since condition (9.22) is valid for a circular orbit in the drift velocity field as well, the radii of stable attracting trajectories and the basins of attraction can be determined from Eqs. (9.51) and (9.22). As a result, the radius of the resonant attractor becomes a nonlinear function of the time delay t in the feedback loop, as shown in Fig. 9.12(b). As under one-point feedback (compare Fig. 9.3), the theoretical predictions are violated for a small attractor radius R < ta 0.2A. In this case the amplitude of the first Fourier component of the feedback signal practically vanishes, and entrainment or asynchronous attractors are observed instead of the resonance attractor [47]. 

It is very important to stress that changes in the geometrical shape of the integration domain can induce bifurcations in the drift velocity field [31, 47, 50, 52]. Let us consider, for example, the drift velocity field computed for an elliptical domain with major axis o = 3A and minor axis b = a/1.1. As shown in Fig. 9.13(a), instead of the stable limit cycle of the resonance attractor in the circular domain of radius Rg, = 1.5A we have two pairs of fixed points where the drift velocity vanishes. In each pair, one fixed point is a saddle and the other one is a stable node. Depending on the initial conditions, the spiral wave approaches one of the two stable nodes. Trajectories of the spiral center obtained by numerical integration of the Oregonator model (9.1) are in perfect agreement with the predicted drift... 

Fig. 9.13. (a) Drift velocity field obtained for global feedback in an elliptical domain... 

The theoretically predicted destruction of the resonance attractor in response to deviations from the circular shape of the integration domain has been confirmed experimentally within the light-sensitive BZ medium. A spiral wave was exposed to uniform illumination proportional to the total gray level obtained in an elliptical integration domain. Fig. 9.13(b) shows the resonant drift mediated during global feedback control. The spiral wave drifts towards a stable node of the drift velocity field. Close to this fixed point the drift velocity becomes very slow. Thus, the experimentally observed termination of the spiral drift at certain positions in a uniform medium is explained in the framework of the developed theory of feedback-mediated resonant drift. 

It was extremely important to demonstrate that because the phase of the feedback signal depends on the spiral orientation, the direction of feedback-induced resonant drift is determined only by the spiral location and does not depend on its initial orientation. Thus, under feedback control the dynamics of spiral waves can be described by a drift velocity field [47, 53]. 

If a feedback can be considered as a sum of signals taken from different sources, the resulting drift direction is found to be the sum of particular drift vectors induced by each separate source. This superposition principle essentially simplifies the analysis of the drift velocity fields [47, 53]. 

---