Palais coordinates

In [22] the resulting ODEs in the center manifold Ad of a relative equilibrium 5 E(2)n were derived, in these Palais coordinates. Suppressing extra parameters, again, the result is... 

Fig. 3.4. The construction of local Palais coordinates (g,v) for a tubular neighborhood of a group orbit SE 2)ut.

How to interpret systems (3.11), (3.12) First of all, let us interpret the Palais coordinates g,v) = e ° ,z,v) e SE 2) x V themselves. Obviously, the definition of Palais coordinates implies that elements u, U2 in the same group orbit have the same component v. Therefore v parametrizes the shape of spatial profiles, in our setting. Indeed, spatial profiles ui,U2 G X° which have the same u-component differ only in g, viz. by a rotation and translation of their spatial profiles ... 

Thus the tip position z may indeed be considered as the translation component z of the Palais coordinates, directly, up to a coordinate transformation (3.16). Let us calculate, briefly, the resulting skew product ODE component for z ... 

For illustration purposes, let us now interpret meandering and drift in terms of the Palais coordinates (e ,z,t>). First note that the relative equilibrium w = 0, which corresponds to the rigidly rotating spiral, becomes an equilibrium 99(0) = 0 of the shape ODE v = ip v) in (3.12). Indeed the group orbit SE 2)u, which corresponds to one and the same shape variable w = 0, contains the time orbit of the rigidly rotating spiral u. Therefore... 

Under the spectral assumptions of section 3.2.3, theorem 1, it is now possible to reduce the perturbed dynamics to a three-dimensional center manifold which is modeled over the group SE 2) itself. In Palais coordinates... 

As was justified in section 3.2.4, the angle a denotes the phase and. 2 the position of the spiral tip. The Palais section coordinate n U is absent here, because the critical spectrum is now three-dimensional, only, and is accounted for by the three-dimensional group SE 2) itself. Therefore the center manifold M. is a graph over the group coordinates e ° ,z) G SE 2). A rigorous derivation of the reduced equation (3.22) has indeed been achieved in [25, 33], under the assumption that the unperturbed spiral wave n (-) is spectrally stabie with the exception of a triple critical eigenvalue due to symmetry see theorem 1. Note that the nonlinearities -y a,z,s) and h a,z,s) obey the lattice symmetry relic of full Euclidean symmetry, namely... 

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