Fidelity and reversibility in the restricted three body problem

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. Two distinct definitions of Fidelity are given. The asymptotic equivalence of REM and FEM is analytically proved for linear symplectic maps with random perturbations. For a given map, the REM plot provides a picture of the dynamic stability regions in the phase space, very easy to obtain for any kind of perturbation and very simple to implement numerically. The REM and FEM for linear symplectic maps are proved to be asymptotically equivalent. The global error growth follows a power law in the regions of integrable (or quasi integrable) motion and an exponential law in the regions of chaotic motion. We prove that the power law exponent is 3/2 for a generic anisochronous system, but drops down to 1/2 if the system is isochronous. Correspondingly the Fidelity F(t) exhibits an exponential decay and −lnF(t) grows just as the square of the FEM or REM error. The Reversibility Error and Fidelity can be used for a quantitative analysis of dynamical systems and are suited to investigate the transition regions from chaotic to regular motion even for Hamiltonian systems with many degrees of freedom such as the N-body problem.