Improving the precision of noisy oscillators

We consider how the period of an oscillator is affected by white noise, with special attention given to the cases of additive noise and parameter fluctuations. Our treatment is based upon the concepts of isochrons, which extend the notion of the phase of a stable periodic orbit to the basin of attraction of the periodic orbit, and phase response curves, which can be used to understand the geometry of isochrons near the periodic orbit. This includes a derivation of the leading-order effect of noise on the statistics of an oscillator’s period. Several examples are considered in detail, which illustrate the use and validity of the theory, and demonstrate how to improve a noisy oscillator’s precision by appropriately tuning system parameters or operating away from a bifurcation point. It is also shown that appropriately timed impulsive kicks can give further improvements to oscillator precision.