Kolmogorov variation: KAM with knobs <i>(à la Kolmogorov)</i>

Marco Sansottera; Veronica Danesi; Marco Sansottera; Veronica Danesi

doi:10.3934/mine.2023089

Mathematics in Engineering

2023, Volume 5, Issue 5: 1-19. doi: 10.3934/mine.2023089

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Kolmogorov variation: KAM with knobs (à la Kolmogorov)

Marco Sansottera ^{1
,
,},
Veronica Danesi ²

1.
Rocketloop GmbH, Hansaallee 154, 60320 Frankfurt, Germany
2.
Department of Mathematics, University of Rome "Tor Vergata", via della Ricerca Scientifica 1, 00133 Rome, Italy

Received: 04 November 2020 Revised: 01 April 2023 Accepted: 01 April 2023 Published: 15 May 2023

In this paper we reconsider the original Kolmogorov normal form algorithm ^[26] 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.
- Kolmogorov normal form,
- perturbation theory,
- KAM theorem
Citation: Marco Sansottera, Veronica Danesi. Kolmogorov variation: KAM with knobs (à la Kolmogorov)[J]. Mathematics in Engineering, 2023, 5(5): 1-19. doi: 10.3934/mine.2023089

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Abstract

In this paper we reconsider the original Kolmogorov normal form algorithm ^[26] 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.

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