Applications of SAGBI-bases in dynamics

The classical reduction techniques of bifurcation theory, Liapunov–Schmidt reduction and centre manifold reduction, are investigated where symmetry is present. The symmetry is given by the action of a finite or continuous group. The symmetry is exploited systematically by using the algebraic structure of the module of equivariant polynomial tuples. We generalize the concept of SAGBI-bases to module-SAGBI basis and explain how to use this concept within the two reduction techniques. Examples illustrate the theoretical results. In particular the reduction onto centre manifold is performed for the Taylor–Couette problem with SO(2)×O(2)-symmetry.