NORMAL-FORM FOR GENERALIZED HOPF-BIFURCATION WITH NONSEMISIMPLE 1 1 RESONANCE/

The primary result of this research is the derivation of an explicit f ormula for the Poincare-Birkhoff normal form of the generalized Hopf b ifurcation with non-semisimple 1 : 1 resonance. The classical nonuniqu eness of the normal form is resolved by the choice of complementary sp ace which yields a unique equivariant normal form. The 4 leading compl ex constants in the normal form are calculated in terms of the origina l coefficients of both the quadratic and cubic nonlinearities by two d ifferent algorithms. In addition, the universal unfolding of the degen erate linear operator is explicitly determined. The dominant normal fo rms are then obtained by rescaling the variables. Finally, the methods of averaging and normal forms are compared. It is shown that the domi nant terms of the equivariant normal form are, indeed, the same as tho se of the averaged equations with a particular choice for the constant of integration.