Gevrey estimates for asymptotic expansions of tori in weakly dissipative systems

TL;DR

This paper proves that the formal asymptotic expansions for invariant tori in weakly dissipative systems satisfy Gevrey estimates, with bounds depending on the Diophantine condition and dissipation order, using a modified Newton method.

Contribution

It introduces a novel approach to establish Gevrey bounds for asymptotic expansions in dissipative KAM systems using a modified Newton method.

Findings

Gevrey estimates are proven for the asymptotic expansions.

The bounds depend on Diophantine conditions and dissipation order.

The method provides coefficient estimates without requiring analyticity.

Abstract

We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift. It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asymptotic expansions for the quasiperiodic solutions and the counterterm satisfy Gevrey estimates. That is, the $n$-th term of the expansion is bounded by a power of $n!$. The Gevrey class (the power of $n!$) depends only on the Diophantine condition of the frequency and the order of the friction coefficient in powers of the perturbative parameter. The method of proof we introduce may be of interest beyond the problem considered here. We consider a modified Newton method in a space of power expansions. As it is custumary in KAM theory, each step of the method is estimated in a smaller domain. In contrast with the KAM results, the domains where we control the Newton method shrink very fast and the Newton method does not prove that the solutions are analytic. On the other hand, by examining carefully the process, we can obtain estimates on the coefficients of the expansions and conclude the series are Gevrey.