Noise-induced switching in an oscillator with pulse delayed feedback: a discrete stochastic modelling approach

We study the dynamics of an oscillatory system with pulse delayed feedback and noise of two types: i) phase noise acting on the oscillator and ii) stochastic fluctuations of the feedback delay. Using an event-based approach, we reduce the system dynamics to a stochastic discrete map. For weak noise we find that the oscillator fluctuates around a deterministic state, and we derive an autoregressive model describing the system dynamics. For stronger noise the oscillator demonstrates noise-induced switching between various deterministic states; our theory provides a good estimate of the switching statistics in the linear limit. We show that the robustness of the system towards this switching is strikingly different depending on the type of noise. We compare the analytical results for linear coupling to numerical simulations of nonlinear coupling, and find that the linear model also provides a qualitative explanation for the differences in robustness to both types of noise. Moreover, phase noise drives the system towards higher frequencies, while stochastic delays do not, and we relate this effect to our theoretical results.