Almost sure asymptotic stability of an oscillator with delay feedback when excited by finite-state Markov noise

An oscillator of the form q¨(t)+2ζq˙(t)+q(t)=−κ[q(t)−q(t−r)] is unstable when the strength of the feedback (κκ) is greater than a critical value (κcκc). Oscillations of constant amplitude persist when κ=κcκ=κc. We study the almost-sure asymptotic stability of the oscillator when κ=κcκ=κc and the system is excited by a two-state Markov noise. For small intensity noise, we construct an asymptotic expansion for the maximal Lyapunov exponent.

► First non-zero term in the asymptotic expansion of the maximal Lyapunov exponent. ► Oscillator with delay feedback can be stabilized by multiplicative noise. ► Chatter suppression in machining using random perturbations of structural parameters.