Shadowing unstable orbits of the Sitnikov elliptic 3-body problem

TL;DR

This paper demonstrates that numerical shadowing can effectively track unstable orbits in the Sitnikov 3-body problem over long durations, highlighting phase space stretching as a key factor in shadowing failure.

Contribution

It shows that unstable orbits in the Sitnikov problem can be shadowed for extended periods, and identifies phase space stretching near escape regions as a cause of shadowing failure.

Findings

Unstable orbits can be shadowed for long durations

Phase space stretching causes shadowing refinement failure

Numerical shadowing is effective in the Sitnikov problem

Abstract

Errors in numerical simulations of gravitating systems can be magnified exponentially over short periods of time. Numerical shadowing provides a way of demonstrating that the dynamics represented by numerical simulations are representative of true dynamics. Using the Sitnikov Problem as an example, it is demonstrated that unstable orbits of the 3-body problem can be shadowed for long periods of time. In addition, it is shown that the stretching of phase space near escape and capture regions is a cause for the failure of the shadowing refinement procedure.