On the convergence of flow map parameterization methods for whiskered tori in quasi-periodic Hamiltonian systems

TL;DR

This paper establishes an a-posteriori theorem for the existence of partly hyperbolic invariant tori in analytic Hamiltonian systems using a convergent KAM iterative scheme within the parameterization method, including explicit constants for computational verification.

Contribution

It provides a rigorous proof framework for the existence of invariant tori in Hamiltonian systems, with explicit constants for computer-assisted proofs, advancing the understanding of quasi-periodic dynamics.

Findings

Proved convergence of a KAM scheme for invariant tori

Derived conditions for analytic parameterizations in smaller complex strips

Provided explicit constants for computer-assisted validation

Abstract

In this work, we obtain an a-posteriori theorem for the existence of partly hyperbolic invariant tori in analytic Hamiltonian systems: autonomous, periodic, and quasi-periodic. The method of proof is based on the convergence of a KAM iterative scheme to solve the invariance equations of tori and their invariant bundles under the framework of the parameterization method. Starting from parameterizations analytic in a complex strip and satisfying their invariance equations approximatly, we derive conditions for the existence of analytic parameterizations in a smaller strip satisfying the invariance equations exactly. The proof relies on the careful treatment of the analyticity loss with each iterative step and on the control of geometric properties of symplectic flavour. We also provide all the necessary explicit constants to perform computer assisted proofs.