A note on the calculation of paths of Hopf bifurcations (Q1825601)

A two-parameter nonlinear system is considered. The classical extended system for Hopf bifurcation points is analyzed. Namely, the system is shown to be \(Z_ 2\)-equivariant. Under certain nondegeneracy conditions, a Bogdanov-Takens bifurcation point is shown to be a symmetry-breaking bifurcation point (pitchfork) of this extended system. Moreover, the symmetric and asymmetric branches parametrize locally the folds and Hopf bifurcation points, respectively, in a neighbourhood of the above pitchfork. The \(Z_ 2\)-equivariance reflects the conjugacy of complex eigenvalues. Numerical applications are hinted.