Resonance attractor

Resonance attractor for spiral waves subjected to one-channel feedback 252... 

Fig. 9.3. Radii of resonance attractors (diamonds) determined experimentally for meandering spiral waves in the light-sensitive BZ medium vs the time delay in the feedback loop. The dashed lines show the theoretical predictions obtained from Eq. (9.23), the solid lines are the boundaries of the basins of attraction according to Eq. (9.24). Radii of observed entrainment and asynchronous attractors are shown by triangles and squares, respectively [47].

Application of one-channel feedback control to spiral waves in the light-sensitive BZ system allows to observe the discrete set of stable resonant attractors experimentally [21, 30, 43, 46]. Note, that Eq. (9.23) for the radius of the resonance attractor contains only one medium dependent parameter (p, which specifies the direction of the resonance drift. To avoid a rather complicated experimental procedure to determine this value, the obtained experimental data were fitted to the theoretically predicted linear dependence (9.23) using p = —0.31. The results are shown in Fig. 9.3 by dashed lines. Then, the boundaries of the basin of attraction were specified in accordance with Eq. (9.24) (solid lines in Fig. 9.3). 

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 ... 

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... 

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... 

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. 

Finally, we like to point out that our results can explain the observations that 0-H BDEs in phenols correlate with ap+. Because of the direct conjugation between the oxygen lone pair and the substituent, the polar stabilization of the phenol can be expected to follow a linear relationship with Op" rather than with ap+. This is also consistent with our computed APSE which correlates linearly with Op with a correlation coefficient of 0.984. Since the Up and the CTp+ scales differ in that Gp+ predicts much larger substituent effects for resonance donors (e.g. 0CH3 oh and NH2) and relatively smaller substituent effects for resonance attractors (e.g. CN and NO2), the overall relationship between ABDE and electron donating substituents. Thus to understand the substituent effects on the 0-H... 

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

In regard to the subject of temporal structures we discuss briefly the generation of multiple attractors by means of appropriate external perturbations in oscillatory chemical reactions [12], resonance effects [13], and the possibility of the control of distribution of dissipation in such systems [14]. 

Volume 37 Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns... 

In such a case we have the so-called soft loss of stability. The newly established regime inside the attracting spot may be either a new equilibrium state, a periodic trajectory, a non-resonant torus, or even a strange attractor (a situation generally referred as instant chaos). The latter option is possible when Oeo bas three zero eigenvalues (see [18], or [129] for systems with symmetry). 

The multiplicity of the disynaptic basin V(A,B) is not automatically correlated to an equal multiplicity of the A-B bond. On the one hand, the location of the attractors should be consistent with the point symmetry and therefore the disynaptic basins multiplicity can be explained by symmetry consideratitms rather than by chemical arguments. On the other hand, conventional multiple bonds are often limit resonance stmctures which are not always the dominant ones. The bond multiplicity QTAIM and ELF criteria were discussed by Qiesnut [101] who concluded that the measures are dependent on the nature of the AB pair. 

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