Geodesic
A NOTE ON HOPF BIFURCATION WITH DIHEDRAL GROUP SYMMETRY
Glasgow mathematical journal, Tome 48 (2006) no. 1, pp. 41-51

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DOI

We consider the standard action of the dihedral group $\bf{D}_n$ of order $2n$ on $\bf{C}$. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on $\bf{C} \oplus \bf{C}$. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series, M. Golubitsky and J. Guckenheimer, eds., Contemporary Mathematics 46, Am. Math. Soc., Providence, R.I. 1986, 131–173) and van Gils and Valkering (Hopf bifurcation and symmetry: standing and travelling waves in a circular chain. Japan J. Appl. Math.3, 207–222, 1986) prove the generic existence of three branches of periodic solutions, up to conjugacy, in systems of ordinary differential equations with $\bf{D}_n$-symmetry, depending on one real parameter, that present Hopf bifurcation. These solutions are found by using the Equivariant Hopf Theorem. We prove that generically, when $n\neq 4$ and assuming Birkhoff normal form, these are the only branches of periodic solutions that bifurcate from the trivial solution.
DOI : 10.1017/S0017089505002855
Mots-clés : 37G40, 34C23, 34C25 
DIAS, ANA PAULA S.; PAIVA, RUI C. A NOTE ON HOPF BIFURCATION WITH DIHEDRAL GROUP SYMMETRY. Glasgow mathematical journal, Tome 48 (2006) no. 1, pp. 41-51. doi: 10.1017/S0017089505002855
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