# American Institute of Mathematical Sciences

September  2009, 25(3): 823-842. doi: 10.3934/dcds.2009.25.823

## Generating functions for Hopf bifurcation with $S_n$-symmetry

 1 Centro de Matemática da Universidade do Porto (CMUP) and Dep. de Matemática Pur, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal 2 School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, United Kingdom 3 CMUP and Dep. de Matemática Pura, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal

Received  September 2008 Revised  March 2009 Published  August 2009

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 $n$.
Citation: Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $S_n$-symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823
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