Nondegeneracy of Hopf points emanating from a \(Z_ 2\)-symmetry-breaking Takens-Bogdanov point (Q1802406)

In a two-parameter, \(\mathbb{Z}_ 2\)-equivariant dynamical system \(\dot x+f(x,\lambda,\alpha)=0\) \(\mathbb{Z}_ 2\)-symmetry-breaking Takens-Bogdanov bifurcation points \((x_ 0,\lambda_ 0,\alpha_ 0)\) are considered. Nondegeneracy conditions are given such that the symmetric and asymmetric Hopf points in the neighborhood of \((x_ 0,\lambda_ 0,\alpha_ 0)\) (defined by the existence of imaginary eigenvalues of the Jacobians) are Hopf bifurcation points w.r.t. \(\lambda\) in the sense that eigenvalue crossing conditions are satisfied which guarantee the bifurcation of periodical orbits.