Effects of time-delayed feedback on chaotic oscillators

We study the effects of time-delayed feedback on chaotic systems where the delay time is both fixed (static case) and varying (dynamic case) in time. For the static case, typical phase coherent and incoherent chaotic oscillators are investigated. Detailed phase diagrams are investigated in the parameter space of feedback gain ([Formula presented]) and delay time ([Formula presented]). Linear stability analysis, by assuming the time-delayed perturbation, varies as [Formula presented] where [Formula presented] is the eigenvalue, gives the boundaries of the stability islands and critical feedback gains ([Formula presented]) for both Rössler oscillators and Lorenz oscillators. We also found that the stability island are found when the delay time is about [Formula presented], where [Formula presented] is an integer and [Formula presented] is the average period of the chaotic oscillator. It is shown that these analytical predictions agree well with the numerical results. For the dynamic case, we investigate Rössler oscillator with periodically modulated delay time. Stability regimes are found for parameter space of feedback gain and modulation frequency in which it was impossible to be stabilized for a fixed delay time. We also trace the detailed routes to the stability near the island boundaries for both cases by investigating bifurcation diagrams.