Flow Map Parameterization Methods for Invariant Tori in Quasi-Periodic Hamiltonian Systems

Abstract

The aim of this paper is to present a method to compute parameterizations of partially hyperbolic invariant tori and their invariant bundles in nonautonomous 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 nonresonant 3-dimensional invariant tori and their invariant bundles around the L1 point.