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On a degenerate Hopf bifurcation

We consider a one-parameter family of differential equations in with m ⩾ 5 and a parameter ε. We assume that for each ε the differential equation has an equilibrium point x(ε), that the Jacobian matrix fx(x(ε), ε) has two pairs of complex eigenvalues εαi ± i(β + εβi) + O(ε2) for i = 1, 2 with α1α2β ≠ 0, and that the other eigenvalues are with ck ≠ 0 for k = 5, …, m. We note that when ε = 0 the eigenvalues of the Jacobian matrix for the equilibrium point x(0) are ±iβ with multiplicity 2, and 0 with multiplicity m − 4. We study the degenerate Hopf bifurcation which takes place in this parameter family at ε = 0.