Stability and bifurcation in a harmonic oscillator with delays

We consider a harmonic oscillator with delays. Linear stability is investigated by analyzing the associated characteristic transcendental equation. The bifurcation analysis of the equation shows that Hopf bifurcation can occur as the delay τ (taken as a parameter) crosses some critical values. The direction and stability of the Hopf bifurcation are considered by using the normal form theory due to Faria and Magalhães. An example is given to explain the results. Numerical simulations support our results.