The simplest attractor in the quotient system is an equilibrium
. In this case, all the
evolution is motion along a closed group orbit, i.e. the spiral moves as a rigid
body, and this movement is rotation:
Let us suppose that the equlibrium 
undergoes
a Hopf
bufurcation. Then the dynamics of the quotient system can be described by
where z is a normal-form complex coordinate on the central
manifold, Z(z) is a vector field on this manifold, and functions
, C(z) and V(z) determine the shape of the central
manifold in the space 
.
System (14, 16) with the last equation left out, is
formally equivalent to Barkley's (1994) model system obtained
from symmetry considerations. To see this, let us introduce new
variables 
and 
by
resolve the system with respect to z to determine the function
z(s,w), where 
(which is a well defined operation if
and 
are nonzero), and then exclude z. We then
obtain
where H(s,w)=1. Meanwhile, system (3) from (Barkley 1994) reads
(the particular choice of the form of F() and G() was for parity purposes, to represent the symmetry due to reflections).
Further bifurcations will normally lead to the increase in the dimensionality of the embedding space of the attractor. For instance, a secondary Hopf bifurcation can give birth to an invariant torus which can subsequently break up leading to dynamical chaos. This scenario would naturally be described in model systems of higher dimensionality. Note that this viewpoint differs from that of Barkley (1994) who tried to reproduce the whole of Winfree's (1991) parametric portrait of the FitzHugh-Nagumo system in full in terms of the same model system, and in particular, to describe the bifurcation of meander into hypermeander.
This ``meander-hypermeander'' bifurcation may be explained as the birth of a
chaotic attractor in the reduced system. To the extent that a chaotic
signal 
has properties analogous to
that of truely stochastic noise, it would be natural to expect that its
time integral R(t) would have properties analogous to that of Wiener
processes, -- i.e. grow at large times in average as 
. Hence, the hypermeander patterns described by
Winfree (1991) and
Nagy-Ungvarai et al. (1993) could be
explained as a Brownian
walk along the symmetry group. The fact that a torus breakup into a chaotic
attractor may occur soon after a secondary Hopf bifurcation
(Afrajmovich & Shilnikov, 1983) is consistent with Winfree's observation of no
other boundaries between the bifurcation lines ``rigid
rotation-meander'' and ``meander-hypermeander''.
