In [1] we have shown how the Euclidean symmetry group of the plane can be exploited to split the analysis of spiral wave dynamics into motion in a reduced non-symmetric `quotient' dynamic system, and drift along the group. For a reaction-diffusion system
with 
, 
,
, the reduced dynamic system is the system
of l PDEs and three finite equations,
for l+3 dynamic variables
,
,
,
,
where the 
are real functions and
. Drift along the group is
described by equations
where 
, and 
and 
are
coordinates along the Euclidean group
, the displacement vector
and rotation angle respectively.
The mathematical idea of this operation is reduction of the dynamics
onto the manifold of group orbits, which is a standard technique for
finite-dimensional systems [2]. Here we used it for PDEs, and
parametrised the orbit manifold by a manifold lying in the phase space
transversal to the orbits. In this particular case, this splitting into
two parts has a simple visual interpretation: the first of Equations
(2) is the reaction-diffusion system in a moving frame of
reference, while the next two say that this frame is chosen so that a
specific point of the spiral wave, naturally associated with the spiral
wave tip, is always located at the origin and the last equation fixes
relative orientation of the spiral tip and of the frame of reference. A
typical choice of functions is:
with some constants 
, 
, which corresponds to the
definition of the tip as intersection of two isolines, in this example
of components 1 and 2, and orientation of the tip defined as that of
the gradient of the first component. Hence, the equations along the
group (3,4) are interpreted as motion equations of
the spiral tip, with R being its complex coordinate and 
its
orientation.
In [1] we have shown that, in particular, this approach provides a simple derivation of Barkley's [3, 4, 5] model system for the bifurcation from simple to compound rotation, recently studied rigorously by Wulff [6] in the original system with symmetry. In our approach, this corresponds simply to the standard Andronov-Hopf bifurcation in the quotient system (2).
The compound two-periodic rotation of spiral wave is not the only type of meander pattern, and more complicated patterns have been reported in literature [7], [8], [9]; Winfree [7] has called this behaviour `hypermeander'. A specific feature of hypermeander is that the spiral tip trajectory appears complicated, and not compact, i.e. it goes outside any prescribed region -- or, at least, has a much larger excursion than the biperiodic meander of the same model with similar values of parameters.
In this paper we discuss the hypothesis put forward in [3, 9, 5, 1] that hypermeander is related to some chaotic behaviour, and, more specifically, a chaotic attractor in the quotient system (2). We show that in this case the spiral tip trajectory is not compact, but walks around the plane, and at large time this walk is analogous to Brownian motion. This prediction is consistent with numerical experiments.
