M. Golubitsky and W.F. Langford
Classification and unfoldings of degenerate Hopf bifurcation
J. Diff. Eqns. 41 (1981) 375-415.
This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that pm i should be simple eigenvalues of the linearized right-hand side at criticality. The the Lyapunov-Schmidt method allows a reduction to a scalar equation G(x,lambda) = 0, where G(-x,lambda) = -G(x,lambda). A definition is given of the codimension of G, and a complete classification is obtained for all problems with codimension le 3, together with the corresponding universal unfoldings. The perturbed bifurcation diagrams are given for the cases with codimension le 2, and for one case with codimension 3; for this last case one of the unfolding parameters is a "modal" parameter, such that the topological codimension equals in fact 2. Formulas are given for the calculation of the Taylor coefficients needed for the application of the results, and finally the results are applied to two simple problems: a model of glycolytic oscillations and the Fitzhugh nerve equations.