Waves rotating spiral

Reaction-diffusion systems encounter difficulties even for the seemingly simple question of mere existence of rigidly rotating spiral waves. With the technically demanding tool of spatial dynamics in the (logarithmic) radial direction, this difficulty has been overcome for small amplitude waves in a celebrated paper by Scheel [66]. Subsequently, interesting consequences have been derived most notably a first classification of possible instabilities of defect dynamics, reiating to core and far-field break-up [71]. 

We briefly comment on the basic steps in the proof of theorem 1. For homogeneous lighting of intermediate strength a rigidly rotating spiral wave solution was assumed to be given by u . The manifold is close to the unperturbed normally hyperbolic center manifold SE(2)u given by the... 

D. Barkley. Euclidean symmetry and the dynamics of rotating spiral waves. Phys. Rev. Lett, 72 164-167, 1994. 

Our attention is focused on the properties of periodic trains formed by kinks or traveling Bloch walls. Our analysis reveals that, depending on the parameters of the oscillatory medium and the spatial period of a train, it can undergo a reversal of its propagation direction [19]. We show how this phenomenon can be used to design traps for traveling kinks and Bloch walls. Furthermore, we find that a new kind of patterns - twisted rotated spiral waves - exist in oscillatory media under the conditions of front propagation reversal. 

The very first attempt to construct a simplified kinematical description of a rotating spiral wave has been done in the classical paper of N. Wiener and A. Rosenblueth [33]. This description is based on the assumption that wave fronts propagate in a uniform and isotropic medium with equal velocity from any stimulated points into a region where the medium is in the rest state. Due to Huygens principle, successive wave fronts are perpendicular to a system of rays which represent the position which may be assumed by stretched cords starting from the stimulated point. The back of the wave is another curve of the same form, which follows the wave front at a fixed distance Ag measured along these rays. 

Fig. 9.1. Different representations of a rigidly rotating spiral wave, (a) Involute of a hole, (b) Solution of the kinematical equation with linear velocity-curvature relationship, (c) Archimedean spiral, (d) Superposition of the three wave fronts where the dotted, dashed and solid lines correspond to (a), (b), and (c), respectively. Far from the rotation center the fronts practically coincide.

The main disadvantage of this kinematical construction is that it corresponds to spiral waves pinned to a hole but fails to describe freely rotating spiral waves in uniform excitable media which are frequently observed both experimentally as well as in numerical simulations of the underlying reaction-diffusion equations. 

A more elaborated kinematical description of a freely rotating spiral wave in a uniform medium is based on the assumption that the normal velocity c of a curved wave front is not a constant, but depends on its curvature [34]. The simplest approximation of this relationship is a linear... 

For a freely rotating spiral, wave front and wave back should coincide at one site, called phase change point [38], see point q in Fig. 9.1(b). In q the normal velocity of the wave front vanishes. This point lies at the shortest distance to the rotation center O. At another interesting point, Q, the radial direction is tangent to the isoconcentration line. 

To determine the shape of a rigidly rotating spiral wave it is convenient to specify the curvature K as a function of the arc length s. A purely geometrical consideration shows that the function K — K[s), the so-called natural representation of a curve, satisfies the following integro-differential equation [39] ... 

Recalling the phenomenon of hard self-excitation in the Oregonator (Chapter III), we have every reason to suspect that, besides a stable homogeneous resting state, there exist stable rotating spiral waves. Furthermore, knowing the 27T/C0q-per iodic solution x(t) of the reaction equations as well as we do, we can immediately describe the spiral wave solution far from the origin ... 

Figure 11.1 Rotating spiral waves during the electrodeposition of silver-antimony alloy. The characteristic wavelength (i.e., the pitch) and the rotation period of the spiral waves are 10 lm and 10 s, respectively. Reprinted with permission from Ref [11].

The phase description can explain expanding target patterns in reaction-diffusion systems. The same method, however, breaks down for rotating spiral waves because of a phase singularity involved. The Ginzburg-Landau equation is then invoked. 

Reaction-diffusion systems are expected to show spatio-temporal chaos in various circumstances. A few specific cases will be discussed. They include the turbuhzation of uniform oscillations, of propagating wave fronts and of rotating spiral waves. 

It is expected that rotating spiral waves obtained for the Ginzburg-Landau equation become unstable and turbulent if l + CiC2<0. This is because spiral waves in general behave asymptotically as plane waves far from the core, and under the above condition no plane waves can remain stable. However, we are not much interested in this kind of turbulence in the present section, but we are more interested in the sort of turbulence which would be caused by the very existence of the phase singularity in the core. 

Cohen, D. S., Neu, J. C., Rosales, R. R. (1978) Rotating spiral wave solutions of reaction-diffusion equations. SIAM J. Appl. Math. 35, 536... 

Koga, S. (1982) Rotating spiral waves in reaction-diffusion systems - Phase singularities and multiarmed waves. Prog. Theor. Phys. 67, 164... 

In this section we derive the analytical solutions for a spiral wave steadily rotating around a circular obstacle and for a free spiral wave and investigate their stability. It was found in Section 2, following Wiener and Rosenblueth [6], that a steadily rotating spiral wave is an involute of the obstacle s boundary and approaches an Archimedean spiral far from it. However, the conditions of applicability of the WR approximation break down near the tip of a steadily rotating spiral because the curvature diverges there. Therefore we must use... 

In this section we consider the typical behaviour of spiral waves in weakly excitable media whose properties vary in time or in space. It is assumed that the variations of these properties are small enough and hence a perturbation theory could be used to determine responses of steadily rotating spiral waves to periodic temporal modulation or to weak spatial gradients. Note that the solution for a steadily rotating spiral wave is invariant in respect to translations and rotations any rotation of the spiral and any shift of its centre yield again a valid solution. Therefore, the respective components of the perturbations are not damped. 

Since the gradient is weak its effect can again be calculated in the perturbation theory. In the zeroth order of the perturbation theory we have a steadily rotating spiral wave with a tip performing uniform rotation around a core of radius Rq at an angular velocity uq given by (33). Hence, its coordinate Xq changes with time as... 

We consider further the phenomenon of the breakup in rotating spiral waves. Our analysis is restricted to the spiral waves which are pinned on the obstacles. The numerical simulations performed in [41] have revealed that, as the radius R of the obstacle reaches some critical value, the wave breaks at the border of the obstacle and a free tip is produced. After a transient, the steady rotation of the free tip around a circular core of a larger radius sets in. 

The critical curvature kd for the breakup of a solitary propagating excitation wave is higher (cf. Figure 10) than the critical curvature kc which sets at the tip of the free rigidly rotating spiral wave. Therefore, for instance, the radius Rq of the core of the free spiral is larger than the minimal radius R ... 

If the curvature dependence of the normal propagation velocity is taken into account, a similar solution for the freely rotating spiral wave could be constructed. 

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