M. Golubitsky and W. 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.