On a degenerate Hopf bifurcation

We consider a one-parameter family of C3 differential equations ẋ = f (x, ε) in ℝm 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 α1α2? ≠ 0, and that the other eigenvalues are εck + O(ε2) ∈ ℝ 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. © 2010 IOP Publishing Ltd.