In this paper, we have proposed a way to study systematically spiral wave meander. The idea starts from a group-theoretic construct using abstraction of the manifold of group orbits, and leads to a system of equations (11, 12, 14) which has a clear interpretation and is suitable both for a theoretical analysis and for simulation. As a partial case, this technique results in a model system for the bifurcation from rigid rotating to biperiodic meander, which is formally equivalent to the previously proposed model system (Barkley 1994). However, here we have explicitly derived, this system, and the construct described is a general approach to treat a broader variety of related problems, e.g. the problem of hypermeander.
There are still someopen questions. The reduced dynamical system defined by (11,12) lacks the symmetry of the original one, so we have assumed that it can be treated as a `generic' system, and all results of the dynamical system theory without symmetry can be used. However, this system is defined in an unusual way, and an accurate mathematical consideration is required in particular cases. For the case of the Hopf-Barkley bifurcation considered above, such a rigorous consideration within the approach based on the Lyapunov-Schmidt reduction has been made by Wulff (1996a, b). It is interesting to mention in this respect that, as Barkley (1994) has pointed out, in the cases considered so far, the bifurcation to meander happens supercritically, while in generic systems both supercritic and subcritic cases are `equally possible'.
Another open question is related to the alternative theory reducing description of perturbed spiral wave dynamics to ODEs (see e.g. (Biktashev & Holden 1995), based on another basic mathematical idea, that of central (inertial) manifold. As it was discussed in (Biktashev & Holden 1995), the applicability of that approach depends upon the solvability of eigenvalue problems for the adjoint linearized operator in spaces of functions rapidly decaying far from the spiral core; the physical interpretation of this condition is that the spiral wave is sensitive only to perturbations located near the core. In the context of the theory of (Biktashev & Holden 1995), this property can be considered as definitive for `proper' spiral waves. However, in the present paper, the conditions of transversality (11) or, more generally, (3) seem fairly generic, and the coherent low-dimensional behaviour considered in Sec. 6 is completely due to the vicinity of bifurcation. A natural interpretation is that meander patterns like those observed in spiral waves may be much more widespread than observed so far.
Combination of the two theories can be performed as either as generalization of (Biktashev & Holden 1995) for meandering spirals, or development of the present theory to account symmetry breaking perturbations. Such a combination can be helpful e.g. in studying meandering spirals under external periodic forcing, which has been studied recently from a phenomenological viewpoint (Mantel & Barkley 1996; Grill et al. 1995).