Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum

Claire Chavaudret; Stefano Marmi

doi:10.3934/dcds.2022009

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2022, Volume 42, Issue 7: 3077-3101. Doi: 10.3934/dcds.2022009

This issue Previous Article Next Article Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation

Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum

Claire Chavaudret^1, , and
Stefano Marmi^2,

1.
IMJ-PRG, Université de Paris, Bâtiment Sophie Germain, Boite Courrier 7012, 8 Place Aurélie Nemours, 75205 Paris Cedex 13, France
2.
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

^*Corresponding author: Claire Chavaudret
^*Corresponding author: Claire Chavaudret

Received: August 2021

Early access: February 2022

Published: July 2022

The second author is supported by the Dynamics and Information Theory Institute at the Scuola Normale Superiore and the MIUR PRIN Project Regular and stochastic behaviour in dynamical systems nr. 2017S35EHN.

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Abstract

One considers a system on $ \mathbb{C}^2 $ close to an invariant curve which can be viewed as a generalization of the semi-standard map to a trigonometric polynomial with many Fourier modes. The radius of convergence of an analytic linearization of the system around the invariant curve is bounded by the exponential of the negative Brjuno sum of $ d\alpha $, where $ d\in \mathbb{N}^* $ and $ \alpha $ is the frequency of the linear part, and the error function is non decreasing with respect to the smallest coefficient of the trigonometric polynomial.

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Mathematics Subject Classification: Primary: 37F50, 39A45; Secondary: 37E99, 39A30, 39A33, 37C05.

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Figure 1. The Brjuno function of $ N\alpha $ for $ \alpha \in (0, 1) $. Red: $ N = 1 $; green: $ N = 2 $; blue: $ N = 3 $; purple: $ N = 4 $

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