Hopf bifurcation and multiple periodic solutions in a damped harmonic oscillator with delayed feedback

The purpose of this manuscript is to study the dynamics of a damped harmonic oscillator with delayed feedback. Different to previous papers, the bifurcation when the linearization at an equilibrium has, for critical value of the parameters, a pair of non-semisimple purely imaginary eigenvalues with geometric multiplicity one and algebraic multiplicity two is considered. By employing the Lyapunov-Schmidt reduction, the criteria for the existence and number of branches of bifurcating periodic solutions are derived. Finally, some numerical simulations are given to support the analytic results.