dynamics of a nonstandard finite-difference scheme for a limit cycle oscillator with delayed feedback

We consider a complex autonomously driven single limit cycle oscillator with delayed feedback. The original model is translated to a two-dimensional system. Through a nonstandard finite-difference (NSFD) scheme we study the dynamics of this resulting system. The stability of the equilibrium of the model is investigated by analyzing the characteristic equation. In the two-dimensional discrete model, we find that there are stability switches on the time delay and Hopf bifurcation when the delay passes a sequence of critical values. Finally, computer simulations are performed to illustrate the theoretical results. And the results show that NSFD scheme is better than the Euler method.