Symmetry and bifurcation near families of solutions |
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Authors: | A Vanderbauwhede |
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Institution: | Rijksuniversiteit Gent, Instituut Theoretische Mechanica, Krijgslaan 271-S9, B-9000 Gent, Belgium |
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Abstract: | Let X, Z and Λ be Banach spaces, M: X × Λ → Z a C1-function, and assume that the equation M(x, λ) = 0 has a family of solutions for λ = 0. In this paper we consider the bifurcation of solutions from this family, for ¦λ¦ small, under the condition that both the unperturbed (λ = 0) and the perturbed (λ ≠ 0) equations have certain symmetry properties. The problem is reduced by the Liapunov-Schmidt method, and the bifurcation equations are solved by a straightforward use of the symmetry. As an application we obtain existence of certain periodic solutions for the undamped Duffing equation, a result recently obtained by Schmitt and Mazzanti using different methods. |
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