Generating functions for Hopf bifurcation with \(S_n\)-symmetry (Q1034173)

The article is devoted to Hopf bifurcation with \(S_n\)-symmetry, which acts by permutation of coordinates. This problem particularly is connected with the behaviour of all-to-all coupled nonlinear oscillators. When studying Hopf bifurcation it is important to find the function of appropriate degree of the Taylor expansion at the bifurcation point of the commuting vector field in normal form. The authors pose the problem how many invariants and equivariants for \(S_n\times S^1\) there are, degree by degree and give the answer by construction generating functions for finite \(n\) using these to find recursive relations. With the aid of these relations the authors determine the number of invariants and equivariants for general \(n\) and show that for sufficiently large \(n\) this number is independent on \(n\). The equivariants of third and fifth degrees are explicitly constructed in the article, that is valid for arbitrary \(n\).