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Generating functions for Hopf bifurcation with $ S_n$-symmetry

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  • Hopf bifurcation in the presence of the symmetric group $ S_n$ (acting naturally by permutation of coordinates) is a problem with relevance to coupled oscillatory systems. To study this bifurcation it is important to construct the Taylor expansion of the equivariant vector field in normal form. We derive generating functions for the numbers of linearly independent invariants and equivariants of any degree, and obtain recurrence relations for these functions. This enables us to determine the number of invariants and equivariants for all $n$, and show that this number is independent of $n$ for sufficiently large $n$. We also explicitly construct the equivariants of degree three and degree five, which are valid for arbitrary $n$.
    Mathematics Subject Classification: 37G40, 37C80, 13A50.


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