Multiple bifurcations in a harmonic oscillator with delayed feedback

In this paper, we consider a harmonic oscillator with delayed feedback. By studying the distribution of the eigenvalues of the characteristic equation, we drive the critical values where Bogdanov–Takens (B–T) bifurcation and zero-Hopf bifurcation occur. The versal unfoldings of the normal forms at the singularity of B–T and a pure imaginary and a zero eigenvalue singularity are given, respectively. Some numerical simulations verify the theoretical results.