Fidelity and Reversibility in the Restricted Three Body Problem

TL;DR

This paper compares the Reversibility Error Method and Fidelity in analyzing how small perturbations affect the dynamics of the restricted three body problem, highlighting their equivalence and differences in measuring system stability.

Contribution

It provides a comparative analysis of REM and Fidelity methods in a physically relevant example, including considerations on round-off versus random perturbations.

Findings

REM does not require unperturbed map computation

Fidelity decay quantifies loss of memory in perturbed systems

Round-off and random perturbations show similar effects in this context

Abstract

We use the Reversibility Error Method and the Fidelity to analyze the global effects of a small perturbation in a non-integrable system. Both methods have already been proposed and used in the literature but the aim of this paper is to compare them in a physically significant example adding some considerations on the equivalence, observed in this case, between round-off and random perturbations. As a paradigmatic example we adopt the restricted planar circular three body problem. The cumulative effect of random perturbations or round-off leads to a divergence of the perturbed orbit from the reference one. Rather than computing the distance of the perturbed orbit from the reference one, after a given number n of iterations, a procedure we name the Forward Error Method (FEM), we measure the distance of the reversed orbit (n periods forward and backward) from the initial point. This approach, that we name Reversibility Error Method (REM), does not require the computation of the unperturbed map. The loss of memory of the perturbed map is quantified by the Fidelity decay rate whose computation requires a statistical average over an invariant region.