http://arxiv.org/abs/1412.0873
We present two methods to analyse the global effects of a small perturbation in a non-integrable Hamiltonian system, choosing as a paradigmatic example the restricted planar three body problem and focusing on its Poincar\`e map for the Jacobi invariant. The cumulative effects on the orbit of random or round-off errors leads to a divergence of the perturbed orbit from the exact one. Rather than computing the distance of the perturbed orbit from the reference one after a given number n of time steps, we measure the distance of the reversed orbit (n time steps forwards and backwards) from the initial point. This approach does not require the knowledge of the unperturbed map. The asymptotic equivalence of the Reversibility Error Method (REM) with the forward error is proved for noisy linear maps, and it is shown to characterize the phase space stability of the perturbed map just as the Lyapunov Characteristic Exponent. A second indicator of chaos, the Cumulative Orbital Elements (COE) method is also presented. The loss of memory of the perturbed map is quantified by the Fidelity and its decay rate. It is found that Fidelity behaves in a different way for randomly perturbed regular and for chaotic orbits. This property, already known for one-dimensional maps, is confirmed for the considered planar three body problem suggesting a possible validity for generic hyperbolic systems.
F. Panichi, L. Ciotti and G. Turchetti
Wed, 3 Dec 14
43/60
Comments: 20 pages, 19 figures