Flow map parameterization methods for invariant tori in Hamiltonian systems

TL;DR

This paper introduces a novel methodology combining flow map, parameterization, and symplectic geometry techniques to efficiently compute invariant tori and their bundles in Hamiltonian systems, with applications to space mission planning.

Contribution

It presents a new approach that reduces computational cost and complexity in finding invariant tori using flow map and parameterization methods within a symplectic framework.

Findings

Successfully applied to invariant tori around librational points in the Restricted Three Body Problem.

Achieved efficient computation of invariant bundles, or whiskers, crucial for dynamical analysis.

Demonstrated the method's effectiveness in space mission design contexts.

Abstract

The goal of this paper is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one (avoiding Poincare maps), parameterization methods reduce the cost of a single step of the derived Newton-like method to be proportional to the cost of a FFT. Symplectic properties lead to some magic cancellations that make the methods work. The multiple shooting version of the methods are applied to the computation of invariant tori and their invariant bundles around librational equilibrium points of the Restricted Three Body Problem. The invariant bundles are the first order approximations of the corresponding invariant manifolds, commonly known as the whiskers, which are very important in the dynamical organization and have important applications in space mission design.