Noise- and delay-induced dynamics near a global bifurcation
A generic model exhibiting a saddle-node bifurcation on a
limit cycle is investigated. The model has served as a prototype example of
excitability, strongly related to the existing global bifurcation, and coherence
resonance, when a stochastic force is added. We extend the system including
time-delayed feedback control according to the Pyragas scheme and study it both in the
presence and absence of Gaussian white noise.
We find that the delay itself is able to create multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems. A bifurcation diagram in the $K-\tau$ plane is given ($K$ being the strength of the control force and $\tau$ the time delay).
Finally, we switch on Gaussian white noise. We compare our results to those of the uncontrolled
system, in particular, the coherence resonance curve and
features of the oscillations and the corresponding power spectra.