Flow map parameterization methods for invariant tori in quasi-periodic Hamiltonian systems

TL;DR

This paper introduces a novel method for computing invariant tori in quasi-periodic Hamiltonian systems, extending flow map parameterization techniques to handle non-autonomous, quasi-periodic dynamics with applications to celestial mechanics.

Contribution

It generalizes flow map parameterization methods to the quasi-periodic setting and introduces fiberwise isotropic tori concepts for invariant tori computation.

Findings

Successfully computed non-resonant 3D invariant tori in the Elliptic Restricted Three Body Problem.

Demonstrated the existence of invariant bundles around the L1 point.

Validated the method's effectiveness in a complex celestial mechanics scenario.

Abstract

The purpose of this paper is to present a method to compute parameterizations of invariant tori and bundles in non-autonomous quasi-periodic Hamiltonian systems. We generalize flow map parameterization methods to the quasi-periodic setting. To this end, we introduce the notion of fiberwise isotropic tori and sketch definitions and results on fiberwise symplectic deformations and their moment maps. These constructs are vital to work in a suitable setting and lead to the proofs of magic cancellations that guarantee the existence of solutions of cohomological equations. We apply our algorithms in the Elliptic Restricted Three Body Problem and compute non-resonant 3-dimensional invariant tori and their invariant bundles around the L1 point.