Asymptotic behaviour of the confidence region in orbit determination for hyperbolic maps with a parameter

TL;DR

This paper studies how confidence regions in orbit determination behave asymptotically for hyperbolic maps, revealing different decay rates in chaotic versus regular scenarios and improving existing theoretical results.

Contribution

It provides new conditions for the decay rate of confidence ellipsoids in chaotic orbit determination, extending previous work and applying findings to well-known hyperbolic maps.

Findings

Uncertainties decay at different rates in chaotic scenarios depending on parameters.

In regular cases, decay rates are uniform regardless of parameters.

Improved theoretical bounds align with numerical experiments.

Abstract

When dealing with an orbit determination problem, uncertainties naturally arise from intrinsic errors related to observation devices and approximation models. Following the least squares method and applying approximation schemes such as the differential correction, uncertainties can be geometrically summarized in confidence regions and estimated by confidence ellipsoids. We investigate the asymptotic behaviour of the confidence ellipsoids while the number of observations and the time span over which they are performed simultaneously increase. Numerical evidences suggest that, in the chaotic scenario, the uncertainties decay at different rates whether the orbit determination is set up to recover the initial conditions alone or along with a dynamical or kinematical parameter, while in the regular case there is no distinction. We show how to improve some of the results in \cite{maro.bonanno}, providing conditions that imply a non-faster-than-polynomial rate of decay in the chaotic case with the parameter, in accordance with the numerical experiments. We also apply these findings to well known examples of chaotic maps, such as piecewise expanding maps of the unit interval or affine hyperbolic toral transformations. We also discuss the applicability to intermittent maps.