Competition of Noise Sources in Systems with Delay: The Role of Multiple Time Scales

We study the combined effect of different noise sources on systems with delay. In a model for machine tool vibrations, specifically regenerative chatter, variation of material parameters or external (additive) noise can amplify vibrations via coherence resonance, while random variation of delay can suppress these vibrations. A key feature is the interplay of the noise with several different inherent time scales in the system: the short period of the intrinsic oscillations, the long time scale of their envelope, and the intermediate time interval for the variation of the delay. Optimizing the suppression of vibrations relies on understanding how these time scales and the bifurcation structure influence the combined effect of the noise sources. Furthermore, this multiple-scales viewpoint provides predictions for related types of stochastic dynamics in other systems with delay.