By Sierpinski W.

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R acts in V by T,p, S and Rep. 23) lies on it. 23) on Mil is represented by an ordinary differential equation, which can be parameterized by X E Po ('): dX ill = F(/1, X) in V. )] commutes with the representations of T,p, S, and Rep on V. ) is r-equivariant. Remark 3. This reduction technique to an ODE should not be confused with the Lyapunov-Schmidt decomposition, which also allows the reduction of the bifurcation problem to a finite-dimensional equation. However the L-S reduction is only valid for the computation of branches of steady-state or time-periodic solutions, and is not suitable, in general, for the analysis of their stability.

The same S-invariance leads immediately to the second part of the claim. 3a-c we illustrate these properties. 3(c) show a picture of the projection of the velocity field on a meridian plane. Notice that T 7I / n Uo is also a Taylor flow invariant under symmetry S. 3(b)). III. 2 Spirals and ribbons III. 2. 1 The Hopf bifurcation with O(2}-symmetry We suppose now that the operator Lo has a pair of critical purely imaginary eigenvalues ±iwo. 33) that these eigenvalues are double, that the eigenvectors associated with iwo have the form (1(r,(),z) = U(r)e i (m8+azl, (2(r,(),z) = S(1(r,(),z) = SU(r)e i (m8-az), and that the eigenvectors associated with -iwo are the conjugates of (1 and (2.

Hence this is a codimension-two problem of bifurcation. 29). Under the foregoing hypothesis, these eigenvalues depend on P and v, and their leading part is easy to compute. )2. 3.

### Algebre des ensembles by Sierpinski W.

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