Spiral attractor

An example of such a set is the Lorenz attractor which occurs in a variety of models. The wild spiral attractor [153] is another fascinating example. ... 

At first the trajectory seems to be tracing out a strange attractor, but eventually it stays on the right and spirals down toward the stable fixed point (Recall that both C and C are still stable at r = 21. jThetimeseriesof y vs. t shows the same result an initially erratic solution ultimately damps down to equilibrium (Figure 9.5.3). 

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

These stable orbits represent the attractors of the spiral drift under one-channel feedback. The basins of attraction are bounded by unstable orbits, which correspond to n = 2m + 1 and have radii... 

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 ... 

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. 

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. 

These steps are represented by the index n in Table 5.4. Each value of n represents an allowed orbital distance for a satellite from its parent attractor. The planets have indices of Neptune(O), Uranus(2), Saturn(6), Jupiter(9), Asteroids(12), Mars(15), Earth(18), Venus(21) and Mercury(24). Because of the self-similar symmetry of the golden spiral this progression can be continued indefinitely on a continuously increasing scale. 

By integrating particular examples, it is discovered that the cycles in Eq. (22) can be of two types, unstable cycles, in which the trajectories spiral into an intersection point of the threshold axes, and stable cycles, in which there is a stable limit cycle attractor in concentration space. Unstable oscillations are found for Eq. (22) in structure IV, Fig. 3, and stable limit cycle oscillations are found for Eq. (22) in Fig. 4 (see Section V.3). It is not known whether other types of asymptotic behavior besides extremal steady states and stable and unstable cycles are possible in Eq. (22). 

The situation is reminiscent of Rayleigh-Benard convection in isotropic fluids where stable roll attractors apparently compete with complex patterns, spiral defect turbulence [115-117], It has been shown very recently that if anisotropy is introduced into this system by inclining the convection cell, a normal roll pattern with dislocation defect turbulence occurs, which looks quite similar to patterns observed in EHC [118]. 

The spiral-like shape of this attractor follows from the shape of homoclinic loops to a saddle-focus (2, 1) which appear to form its skeleton. Its wildness is due to the simultaneous existence of saddle periodic orbits of different topological type and both rough and non-rough Poincare homoclinic orbits. 

The significance of higher degeneracies (starting from codimension three) in the linear part is that the effective normal forms become three-dimensional, and may, as a result, exhibit complex dynamics, the so-called instant chaos, even in the normal form itself. Such examples include the normal forms for a bifurcation of an equilibrium state with a triplet of zero characteristic exponents, and a complete or incomplete Jordan block, in which there may be a spiral strange attractor [18], or a Lorenz attractor [129], respectively (the latter case requires an additional symmetry). Since we will focus our considerations only on simple dynamics, we do not include these topics in this book. 

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