Spiral waves are rotating waves of reaction-diffusion equations on the plane. In this Note, a center-manifold reduction for the dynamics of spiral waves is presented. Bifurcations of rigidly-rotating spiral waves are then described by ordinary differential equations, which are equivariant under the (special) Euclidean group SE(2). Several difficulties arise in this analysis because SE(2) is not compact and does not induce a strongly continuous group action on the underlying function space.
|Translated title of the contribution||Center-manifold reduction for spiral waves|
|Number of pages||6|
|Journal||Comptes Rendus de l'Academie des Sciences - Series I: Mathematics|
|State||Published - Jan 1997|
Bibliographical noteFunding Information:
The main technique for proving theorem 1 is the graph transform applied to (PT(u, I-L) for some large time II’. Hypothesis 3 guaranteest he existence of smooth, equivariant center-unstable and stable bundles over the group orbit SE(Z)YL* using, for instance, Dunford integrals. In order for the graph transform to work. the vector field has to be modified in the central directions only. If the modified vector field were smooth and equivariant, a standard argument would show that the resulting invariant center-unstable manifold would also be equivariant. Hypothesis 1 implies that the rotations To,, act smoothly on the center-unstable bundle. Indeed, To,-~, t DGt(~u,,p,) is a bounded group on the finite-dimensional space EL”. Because of local compactness of EztL and hypothesis 1, this group is actually continuous in f. Thus, it is in fact analytic as it is given by eUt for some matrix B. We shall emphasize that the matrix B determines the linearization D,,y,(O, IL,) of the normal vector field Q~. Also. if Y = C,‘~,,~.,(IwI”w. ”), we have U = L/E,‘, , see (5). Smoothnesso f the action of SE(2) on the center bundle is now used to modify the vector field equivariantly and smoothly near the group orbit. Then an application of the graph transform shows the existence of the center-unstable manifold having the properties stated in theorem 1. Finally, the representation of the flow on the manifold is obtained by considering the lifting of the vector field to the covering space SE(2) x V, of the center-unstable bundle. Here, the covering is defined by ((1,.c p.* II) ++ T,,,q(~~* + ~1). Details will appear in . B. S. : Partially supported by a Feodor-Lynen fellowship of the Alexander von Humboldt foundation. Cl. W. : Partially supported by the Deutsche Forschungsgemeinschaft under grant Fi441/5-2.