Orbit determination for standard-like maps: asymptotic expansion of the confidence region in regular zones

TL;DR

This paper analyzes the asymptotic behavior of confidence regions in orbit determination for generalized standard-like maps, providing analytical insights into the uncertainties in regular zones as observations increase.

Contribution

It offers an analytical description of the confidence region geometry and its asymptotic trend in regular zones for a class of cylinder maps, extending prior numerical findings.

Findings

Confidence regions tend to shrink asymptotically with more observations.

The geometry of the confidence region is characterized in regular zones.

Analytical estimates of uncertainty trends are established for invariant curves.

Abstract

We deal with the orbit determination problem for a class of maps of the cylinder generalizing the Chirikov standard map. The problem consists of determining the initial conditions and other parameters of an orbit from some observations. A solution to this problem goes back to Gauss and leads to the least squares method. Since the observations admit errors, the solution comes with a confidence region describing the uncertainty of the solution itself. We study the behavior of the confidence region in the case of a simultaneous increase of the number of observations and the time span over which they are performed. More precisely, we describe the geometry of the confidence region for solutions in regular zones. We prove an estimate of the trend of the uncertainties in a set of positive measure of the phase space, made of invariant curve. Our result gives an analytical proof of some known numerical evidences.