[Paper Review] Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

Ernest Fontich, Rafael de la Llave+1 more|ArXiv.org|Mar 2, 2009

Quantum chaos and dynamical systems38 references3 citations

TL;DR

This paper establishes the existence of invariant whiskered tori in finite-dimensional symplectic maps and flows using an a posteriori method based on solving a functional equation for invariance. By proving that any sufficiently accurate approximate solution satisfying non-degeneracy conditions is close to a true solution, the approach avoids assumptions of integrability or action-angle variables, enabling the treatment of primary and secondary tori uniformly with general hyperbolic bundles.

ABSTRACT

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an {\sl a posteriori} approach. The proof of the main theorems is based on an iterative method to solve the functional equation. The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant. The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on succesive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.

Motivation & Objective

• To prove the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows.
• To develop a general framework that does not require the system to be close to integrable or expressed in action-angle variables.
• To handle both primary and secondary tori uniformly, including those near rank-1 resonances.
• To establish a robust method applicable to approximate solutions from numerical or formal methods.
• To provide a foundation for extending the method to infinite-dimensional systems.

Proposed method

• The method solves the invariance equation $ F \circ K = K(\theta + \omega) $ for maps and $ \partial_\omega K = J \nabla H(K) $ for flows via an iterative Newton-type scheme.
• It uses a parameterization approach where an approximate solution $ K_0 $ is assumed to satisfy the invariance equation up to a small error $ E_0 $.
• Non-degeneracy conditions—spectral and twist conditions—are imposed on the approximate solution to ensure solvability of the linearized equations.
• The linearized equations are solved separately on the center and hyperbolic subspaces, with small divisors handled via Diophantine frequency assumptions.
• The iterative correction process preserves hyperbolicity and non-degeneracy, ensuring convergence to a true solution.
• The method relies on successive corrections rather than transformation theory, enabling adaptation to infinite-dimensional settings.

Experimental results

Research questions

• RQ1 Can invariant whiskered tori be constructed in finite-dimensional symplectic systems without assuming integrability or action-angle variables?
• RQ2 How can approximate solutions from numerical or formal methods be rigorously validated as close to true invariant tori?
• RQ3 What non-degeneracy conditions ensure the existence of a true solution near an approximate one in the presence of small divisors?
• RQ4 Can the method handle secondary tori near rank-1 resonances and non-trivial stable/unstable bundles?
• RQ5 What regularity and dependence properties (e.g., analyticity, Lipschitz) can be established for the invariant tori?

Key findings

• If the error $ \|E_0\|_{\rho_0} $ of an approximate solution $ K_0 $ is sufficiently small, specifically $ C\kappa^4\delta^{-4\nu}\|E_0\|_{\rho_0} < 1 $, then a true solution $ K_\infty $ exists in a smaller domain $ \rho_\infty = \rho_0 - 6\delta $.
• The distance between the true solution and the approximate solution satisfies $ \|K_\infty - K_0\|_{\rho_\infty} \leq C\kappa^2\delta^{-2\nu}\|E_0\|_{\rho_0} $, quantifying the convergence rate.
• The method applies to both maps and flows, with results for Hamiltonian systems derived from the general exact symplectic framework.
• The approach does not require the hyperbolic bundles to be trivial or the hyperbolic motion to be constant, allowing for general stable and unstable directions.
• The method ensures the existence of invariant tori even when the system is not close to integrable, enabling treatment of secondary tori and resonant structures.
• The framework supports bootstrap of regularity and provides Lipschitz and analytic dependence on parameters, with explicit measure estimates for the set of frequencies.