Abstract
We develop a normal form theory for parametrized vector fields, respecting their parameter dependences. This parametrized normal form theory (P.N.F.T.) can be applied to the study of bifurcations for systems of evolution equations with the help of the center manifold theory. As an application, we investigate bifurcations of some systems of reaction diffusion equations, considering the diffusion coefficients as the bifurcation parameters. Some detailed bifurcational aspects are given around a doubly degenerate singularity. Under certain conditions, the occurrence of a Hopf bifurcation is shown as a tertiary bifurcation from the trivial solution.
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Kokubu, H. Normal forms for parametrized vector fields and its application to bifurcations of some reaction diffusion equations. Japan J. Appl. Math. 1, 273–297 (1984). https://doi.org/10.1007/BF03167061
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DOI: https://doi.org/10.1007/BF03167061