Rapid and Accurate Methods for Computing Whiskered Tori and their Manifolds in Periodically Perturbed Planar Circular Restricted 3-Body Problems

TL;DR

This paper introduces rapid, accurate computational methods for identifying whiskered tori and their manifolds in periodically perturbed planar circular restricted 3-body problems, enhancing efficiency and applicability.

Contribution

It develops a quasi-Newton method, implements continuation techniques, and computes Fourier-Taylor parameterizations, advancing the numerical analysis of perturbed RTBP systems.

Findings

Improved computational efficiency and accuracy over previous methods

Successful application near resonances in the elliptic RTBP

Effective globalization of stable and unstable manifolds

Abstract

When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology improves on efficiency and accuracy compared to prior studies, and applies to a variety of periodic perturbations. We demonstrate the tools near resonances in the planar elliptic RTBP.