Simultaneous computation of whiskered tori and their whiskers in Hamiltonian systems using flow maps

TL;DR

This paper introduces an efficient algorithm based on the parameterization method for simultaneously computing whiskered tori and their stable and unstable manifolds in Hamiltonian systems, with applications demonstrated in celestial mechanics.

Contribution

The paper develops a novel, efficient algorithm for the simultaneous computation of invariant tori and their whiskers in Hamiltonian systems, enhancing stability and performance.

Findings

Quadratic convergence of the Newton-like method.

Low storage and operation count due to geometric identities.

Successful implementation in the Circular Restricted Three Body Problem.

Abstract

We consider autonomous Hamiltonian systems and present an algorithm to compute at the same time partially hyperbolic invariant tori (whiskered tori), as well as high-order expansions of their stable and unstable manifolds. Such whiskered tori have been shown to be important for transport phenomena in phase space. For instance, by following their invariant manifolds one could obtain zero-cost trajectories in space mission design. We present in detail the case when the (un)stable directions are one-dimensional. The strategy to compute tori and their invariant manifolds is based on the parameterization method. We formulate a functional equation for a parameterization of both the torus and its whiskers expressing that they are invariant. This equation is naturally discretized in Fourier-Taylor series or, equivalently, in a grid of Taylor series. Using a return map, we are reduced to study functions of n - 1 variables where n is the number of degrees of freedom (the phase space is 2n dimensional). Then, we implement a Newton-like method that converges quadratically. They key advantage of our approach is that, using geometric identities coming from the Hamiltonian nature of the problem, the algorithm has small storage requirements and a low operation count per step which is highly efficient. The simultaneous computation of the torus and the whiskers improves the efficiency and the stability of the algorithm. We present implementations and extensive numerical experiments in the Circular Restricted Three Body Problem.