Hopf Bifurcation with Dn-symmetry
نویسندگان
چکیده
The aim of this paper is to study Hopf bifurcation with Dn-symmetry assuming Birkhoff normal form. We consider the standard action of Dn on C. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on C ⊕ 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 Dn-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 6= 4, these are the only branches of periodic solutions that bifurcate from the trivial solution. AMS classification scheme numbers: 37G40 34C23 34C25
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