An Uncertainty Principle for Estimates of Floquet Multipliers

Abstract

We derive a Cram\'er-Rao lower bound for the variance of Floquet multiplier estimates that have been constructed from stable limit cycles perturbed by noise. To do so, we consider perturbed periodic orbits in the plane. We use a periodic autoregressive process to model the intersections of these orbits with cross sections, then passing to the limit of a continuum of sections to obtain a bound that depends on the continuous flow restricted to the (nontrivial) Floquet mode. We compare our bound against the empirical variance of estimates constructed using several cross sections. The section-based estimates are close to being optimal. We posit that the utility of our bound persists in higher dimensions when computed along Floquet modes for real and distinct multipliers. Our bound elucidates some of the empirical observations noted in the literature; e.g., (a) it is the number of cycles (as opposed to the frequency of observations) that drives the variance of estimates to zero, and (b) the estimator variance has a positive lower bound as the noise amplitude tends to zero.