Hopf bifurcation at a double eigenvalue (Q1373021)

Consider the \(n\)-dimensional system of ordinary differential equations \[ dx/dt= f(x,\mu)\tag{\(*\)} \] depending on a real parameter. \(f\) is assumed to be smooth and to satisfy \(f(0,\mu)\equiv 0\;\forall\mu\). Let be \(A(\mu):= f_x(0,\mu)\). The author studies Hopf bifurcation of \((*)\) in case that \(\pm i\) are double eigenvalues of \(A(\mu_0)\). The essential tool is the Lyapunov-Schmidt reduction technique. There is no comparison of the obtained results with other results concerning Hopf bifurcation at a multiple eigenvalue.