The apparently chaotic behaviour like that shown in Fig. 3
looks similar to the well known chaos in Kuramoto-Sivashinsky equation.
This is not a coincidence. If 
in equation (3), then initially planar
curve will remain planar. And if we supply this equation with fourth-order
spatial derivative for regularisation and rewrite it for function, say,
Y(X,t), it will lead to Kuramoto-Sivashinsky equation,
which is quite natural, since the origin of the latter, the short-wave
instability of shape of propagating front, is similar to that of (3).
Note however, that the analogy is not exact since Eq. (3)
is for the period-averaged position of the filament, while the actual evolution
for the filament is more complicated even in this simplest case
.
In the more general case, 
, the behaviour is more complicated still and is essentially three-dimensional.
In hydrodynamics, 3-D turbulence is significantly more complicated phenomenon
than 2-D one. For AW media, as we have mentioned above, some types of instabilities
are possible in 2-D. It would be interesting to compare properties of these
different instabilities and chaos generated by them, to see if 3-D AW turbulence
is also much more complicated than 2-D one. An indirect evidence for that
is in the fact that 3-D turbulence may arise in media which reveal no special
properties in two dimensions (i.e. spiral wave rotate rigidly).
Another interesting analogy with hydrodynamic turbulence is presence of coherent structures. It has been mentioned that in large media and developed AW turbulence, there are repeating motives in the vortices dynamics, like pairs of helical vortices. Existence of coherent structures, including those build up of vortices, is well know in hydrodynamics [Monin, 1994].
The most interesting possible realisation of the 3-D AW turbulence described here is the ventricular fibrillation, a severe life-threatening pathology which occurs as a terminal stage of various cardiac diseases. Despite the long history of the question, phenomenon of fibrillation is not fully understood as yet (which is also true for the hydrodynamic turbulence). The most popular view of the fibrillation is that it involves permanent creation, evolution, multiplication and annihilation of multiple excitation wavelets and micro-reentries, which is the electrophysiological term for the AW vortices. However, the detailed mechanisms of these elementary processes remain unclear. Attempts to explain these in terms of spiral wave evolution in excitable media have been made since the work of Moe et al., [1964], where the immediate cause of the unordered behaviour was random scattering of cellular properties, and [Krinsky, 1968], where the key process was interaction of excitation waves with sharp stepwise tissue inhomogeneities. A more recent discovery possibly relevant to this phenomenon is a 2-D instability of spiral waves, seen in a variant of the FHN model [Panfilov & Holden 1990, 1991], in a model of myocardium [Winfree, 1989; Courtemanche & Winfree 1991] and in a model of Pt-catalysed oxidation of CO [Bär et al., 1994], where intensive meander of the spiral leads to breakup of the radiated wavefronts and thus to generation of new spirals. The 3-D AW turbulence described here suggests another mechanism of fibrillation, different from those described so far in two main points, that it is not stipulated by medium inhomogeneities, and it is essentially 3-dimensional. Negative filament tensions have been observed so far in excitable media with relatively low excitability, and this correlates with the increased likeliness of fibrillation in `fatigued' tissue.