Kolmogorov variation: KAM with knobs (\`a la Kolmogorov)

TL;DR

This paper introduces a variation of the Kolmogorov normal form algorithm that dynamically adjusts frequencies during normalization, allowing for flexible selection of the final invariant torus frequencies and leveraging Newton-Kantorovich convergence.

Contribution

It presents a novel approach to the Kolmogorov algorithm by changing frequencies instead of fixing them, enabling more adaptable invariant torus computations.

Findings

The modified algorithm converges efficiently using Newton-Kantorovich methods.

It allows for selecting final frequencies and determining initial ones post-process.

The approach offers a flexible framework for invariant torus analysis.

Abstract

In this paper we reconsider the original Kolmogorov normal form algorithm with a variation on the handling of the frequencies. At difference with respect to the Kolmogorov approach, we do not keep the frequencies fixed along the normalization procedure. Besides, we select the frequencies of the final invariant torus and determine a posteriori the corresponding starting ones. In particular, we replace the classical translation step with a change of the frequencies. The algorithm is based on the original scheme of Kolmogorov, thus exploiting the fast convergence of the Newton-Kantorovich method.