For our present case of diffusion along 
, application of the
Theorem is not so simple, as the group manifold not only is not a
straight line (it is three-dimensional), but is `curved', due to the
non-commutativity of .
Formally, this is represented by the fact that equations
(3,4) neither have a simple form (8), nor can
be reduced to one. So, the straightforward identification of q of the
Theorem with Euclidean-group coordinates X, Y and 
, and the
semiflow 
with that generated by the dynamical system
(2), does not work. In this case, the tip coordinates
are determined via two indefinite integrals, the
first to find 
from 
, and the second to find
R(t) from c(t) and . Correspondingly, the Theorem
should be applied twice, to each of the indefinite integrals, and the
second application does not succeed due to the possibility of
correlation between and c(t), as they come from the same
dynamic system. Conceivably, this is not just a technical difficulty,
but reflects the essence of the problem, as it is seen from the
following alternative viewpoint.
Let us reduce (1) by the subgroup of translations only. The quotient system is
for dynamic variables w,
and 
, and motion along the group is motion of the
tip,
Note that the `semi-reduced system' (17) is still
invariant under the subgroup of rotations of the Euclidean group,
, represented as simultaneous rotations of vector
and arguments of w.
The subgroup of translations is commutative, the dynamics of X and Y
are separated, and we can identify q of the Theorem with X and Y,
V with 
and 
, and with the semiflow generated by
(17). As the semi-reduced system is still
invariant under 
, its maximal attractor must be
invariant under this group, but can be split into invariant subsets,
each invariant under a subgroup 
.
This subgroup could be either 
itself, in
which case the only invariant subset coincides with the whole maximal
attractor, or 
, 
or the trivial group, and then
there are many invariant subsets.
Let us consider all these cases:
is or ,
m>1, then
the mean value of vanishes, as required by the Theorem, and
the tip undergoes Brownian motion in the plane. As the motion is in
both the x and y directions, the mean squared walking distance is
where 
and 
are parameters of correlation function of each of
the velocities and . This corresponds to diffusion
coefficient 
.
is the trivial
group, then the mean value of the tip velocity 
is
generically non-zero, if averaged over an ergodic component of measure
, corresponding to an invariant subset of the attractor. In
this case, there will be a superposition of directed drift with the
velocity 
and direction depending on the particular trajectory
at the maximal attractor, and a Brownian walk.
In a particular system, either case may take place. In an analogous
problem, that of periodic solutions in systems with finite symmetry
groups, it is known that symmetry subgroup of a periodic solution is a
robust property, in the sense that it changes only at bifurcation
points [12]. Here we deal with continuous groups and chaotic
attractors, and so the question is more complicated. As pointed out
by Mantel and Barkley [13], in the case of quasi-periodic
behaviour in the quotient system, there is directed drift and hence
splitting of the attractor in the semi-reduced system for a set of
parameters which is everywhere dense, as well as the set of non-split
attractors. For the chaotic case considered here, results of Jones and
Parry [14] can be applied. Namely, Theorem 5 of [14]
claims, in certain assumptions, that if the quotient system is uniquely
ergodic, the full system is also uniquely ergodic, for `almost all'
parameters. In our case, assuming unique ergodicity in (2)
and applied to the group , this theorem means that a non-split
attractor in (17) is the typical case, while a split
attractor is the case of measure zero. However, this does not exclude
the possibility of both the cases being everywhere dense.
Note, that the trajectory of the tip would be erratic in either case,
and in practice they may be indistinguishable, especially if the
hypermeander diffusion coefficient is large and the mean
velocity 
is small, or experimental noise or round-off errors are
significant.
To sum up, we see that if the quotient system of the spiral wave has a chaotic attractor, then the trajectory of the tip will not only look complicated, but be non-compact. The long-time asymptotics of tip motion would be either a Brownian motion or a superposition of the Brownian motion and a directed drift, with direction depending on initial conditions and perhaps slowly varying due to experimental noise or roundoff errors.
In either case, a chaotic attractor in the quotient system may explain the corresponding feature of the hypermeandering spiral waves.
