and so f' is bounded only if ^ = w(A^). The first of (1.140) determines f '(m) as [MAm)/D\l/2. We thus have the dispersion relation

which shows how the amplitude at infinity determines the frequency Near r = 0, set

where a0 = 0, substitute into the first of (1.140) and equate powers of r in the usual way. The coefficient of lowest order, namely, rc-2, set equal to zero gives c(c — 1) + c — m2 = 0 ^ c = ±m.

For A(r) to be nonsingular as r ^ 0 we must choose c = m, with which

A(r) ~ a0rm, as r ^ 0, where a0 is an undetermined nonzero constant. The mathematical problem is to determine a0 and ^ so that A(r) and f '(r) remain bounded as r ^ <x>. From (1.137), (1.139) and the last equation, we get the behaviour of u and v near r = 0 as

Koga (1982) studied phase singularities and multi-armed spirals, analytically and numerically, for the X-rn system with

where p > 0. Figure 1.22 shows his computed solutions for 1-armed and 2-armed spirals.

Figure 1.22. Computed (a) 1-armed and (b) 2-armed spiral wave solutions of the k-rn system (1.135) with the k and rn given by (1.145) with f = 1. Zero flux boundary conditions were taken on the square boundary. The shaded region is where u > 0. (From Koga 1982, courtesy of S. Koga)

Figure 1.22. Computed (a) 1-armed and (b) 2-armed spiral wave solutions of the k-rn system (1.135) with the k and rn given by (1.145) with f = 1. Zero flux boundary conditions were taken on the square boundary. The shaded region is where u > 0. (From Koga 1982, courtesy of S. Koga)

The basic starting point to look for solutions is the assumption of the functional form for u and v given by

With A(r) a constant and f(r) a ln r these represent rotating spiral waves as we have shown. Cohen et al. (1978) proved that for a class of k(a) and w(a) the system (1.136) has rotating spiral waves of the form (1.146) which satisfy boundary conditions which asymptote to Archimedian and logarithmic spirals; that is, f ~ cr and f ~ c ln r as r ^œ. Duffy et al. (1980) showed how to reduce a general reaction diffusion system with limit cycle kinetics and unequal diffusion coefficients for u and v, to the case analysed by Cohen et al. (1978).

Kuramoto and Koga (1981) studied numerically the specific k-w system where

where e > 0 and a > 0. With these the system (1.136) becomes wt = (e + ic)w — (a + ib) \ w |2w + DV2w.

We can remove the c-term by setting w ^ weict (algebraically the same as setting c = 0) and then rescale w, t and the space coordinates according to to get the simpler form wt = w - (1 + iß)\ w \2w + V2w,

where i = b/a. The space-independent form of the last equation has a limit cycle solution w = exp (—ifi t).

Kuramoto and Koga (1981) numerically investigated the spiral wave solutions of (1.147) as | i | varies. They found that for small | i | a steadily rotating spiral wave developed, like that in Figure 1.22(a) and of the form (1.144). As | i | was increased these spiral waves became unstable and appeared to become chaotic for larger | i |. Figure 1.23 shows the results for | i | =3.5.

Kuramoto and Koga (1981) suggest that 'phaseless' points, or black holes, such as we discussed in Chapter 9, Volume I, start to appear and cause the chaotic instabilities. Comparing (1.147) with (1.136) we have X = 1 — A2, w = — i A2 and so i is a measure of how strong the local limit cycle frequency depends on the amplitude A. Since A varies with the spatial coordinate r we have a situation akin to an array of coupled, appropriately synchronized, oscillators. As | i | increases the variation in the oscillators increases. Since stable rotating waves require a certain synchrony, increasing the variation in the local 'oscillators' tends to disrupt the synchrony giving rise to phaseless points and hence chaos. Chaos or turbulence in wavefronts in reaction diffusion mechanisms has been considered in detail by Kuramoto (1980); see other references there to this interesting problem of spatial chaos.

To conclude this section let us look at the 1-dimensional analogue of a spiral wave, namely, a pulse which is emitted from the core, situated at the origin, periodically and on alternating sides of the core. If the pulses were emitted symmetrically then we would

Figure 1.23. Temporal development (time T) of chaotic patterns for the k — rn system (1.147) for f = 3.5 and zero flux boundary conditions. (From Kuramoto and Koga 1981, courtesy of Y. Kuramoto)

Figure 1.23. Temporal development (time T) of chaotic patterns for the k — rn system (1.147) for f = 3.5 and zero flux boundary conditions. (From Kuramoto and Koga 1981, courtesy of Y. Kuramoto)

have the analogue of target patterns. Let us consider (1.138) with V2 = d2/dx2, X(a) = 1 - A2 and w(a) = qA2. Now set x ^ D/i' A = A(x), $ = Qt + f(x) to get as the equations for A and f,

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