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

摘要

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.