Computer-assisted proofs of existence of fiberwise hyperbolic invariant tori in quasi-periodic systems via Fourier methods

Alex Haro; Eric Sandin Vidal

doi:10.3934/dcdsb.2024143

Computer-assisted proofs of existence of fiberwise hyperbolic invariant tori in quasi-periodic systems via Fourier methods

Alex Haro^*, Eric Sandin Vidal

^*Corresponding author for this work

Mathematics

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Abstract

The goal of this paper is to provide a methodology to prove the existence of (fiberwise hyperbolic) real-analytic invariant tori in real-analytic quasi-periodic skew-product dynamical systems that present nearly-invariant tori of the same characteristics. The methodology was based on the application of a Newton-Kantorovich theorem whose hypotheses were tested using Fourier analysis methods for a numerical approximation of the parameterization of an invariant torus.

Original language	English
Pages (from-to)	2051-2072
Number of pages	22
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	30
Issue number	6
Early online date	Oct 2024
DOIs	https://doi.org/10.3934/dcdsb.2024143
Publication status	Published - Jun 2025

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© 2025 American Institute of Mathematical Sciences. All rights reserved.

Keywords

computer-assisted proof
Fourier methods
Invariant tori
normal hyperbolicity
quasi-periodically forced system

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10.3934/dcdsb.2024143

Computation_of_invariant_tori__Accepted_original_Final published version, 846 KBLicence: Article 25fa Dutch Copyright Act

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abstract = "The goal of this paper is to provide a methodology to prove the existence of (fiberwise hyperbolic) real-analytic invariant tori in real-analytic quasi-periodic skew-product dynamical systems that present nearly-invariant tori of the same characteristics. The methodology was based on the application of a Newton-Kantorovich theorem whose hypotheses were tested using Fourier analysis methods for a numerical approximation of the parameterization of an invariant torus.",

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AB - The goal of this paper is to provide a methodology to prove the existence of (fiberwise hyperbolic) real-analytic invariant tori in real-analytic quasi-periodic skew-product dynamical systems that present nearly-invariant tori of the same characteristics. The methodology was based on the application of a Newton-Kantorovich theorem whose hypotheses were tested using Fourier analysis methods for a numerical approximation of the parameterization of an invariant torus.

KW - computer-assisted proof

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KW - Invariant tori

KW - normal hyperbolicity

KW - quasi-periodically forced system

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JF - Discrete and Continuous Dynamical Systems - Series B

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ER -

Computer-assisted proofs of existence of fiberwise hyperbolic invariant tori in quasi-periodic systems via Fourier methods

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Keywords

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