Remark on the Hopf Bifurcation Theorem

A simple generalization of the Hopf Bifurcation Theorem for scalar higher order ordinary differential equations is suggested. We study the degenerate case where several roots of the characteristic polynomial cross the imaginary axis at the same point for some value λ₀ of the parameter λ. The main result is that if N₁ roots cross the imaginary axis from the left to the right and N₂ roots cross it from the right to the left, then λ₀ is a Hopf bifurcation point whenever N₁ ≠ N₂. In particular, in the classical Hopf Bifurcation Theorem the numbers N_j are 0 and 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)