Bifurcation analysis in a limit cycle oscillator with delayed feedback

A kind of limit cycle oscillator with delayed feedback is considered. Firstly, the linear stability is investigated. According to the analysis results, the bifurcation diagram is drawn in the parameter plane. It is found that there are stability switches for time delay, and Hopf bifurcations when time delay crosses through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form method and the center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results found.