Asymptotic behaviour of orbit determination for hyperbolic maps

TL;DR

This paper analyzes the asymptotic properties of orbit determination for hyperbolic maps, focusing on how confidence regions evolve with increasing observations and parameters, revealing the impact of including dynamical parameters.

Contribution

It provides a theoretical analysis of the asymptotic behavior of confidence regions in orbit determination, especially regarding the effect of adding dynamical parameters.

Findings

Increased observations lead to shrinking confidence regions.

Including dynamical parameters alters the decay rate of uncertainties.

The geometry of confidence regions depends on parameter inclusion.

Abstract

We deal with the orbit determination problem for hyperbolic maps. The problem consists in determining the initial conditions of an orbit and, eventually, other parameters of the model from some observations. We study the behaviour of the confidence region in the case of 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 the solution, distinguishing whether a parameter is added to the estimate of the initial conditions or not. We prove that the inclusion of a dynamical parameter causes a change in the rate of decay of the uncertainties, as suggested by some known numerical evidences.