April 2023 Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems
Maxime Breden, Maximilian Engel
Author Affiliations +
Ann. Appl. Probab. 33(2): 1252-1294 (April 2023). DOI: 10.1214/22-AAP1841

Abstract

We confirm a long-standing conjecture concerning shear-induced chaos in stochastically perturbed systems exhibiting a Hopf bifurcation. The method of showing the main chaotic property, a positive Lyapunov exponent, is a computer-assisted proof. Using the recently developed theory of conditioned Lyapunov exponents on bounded domains and the modified Furstenberg–Khasminskii formula, the problem boils down to the rigorous computation of eigenfunctions of the Kolmogorov operators describing distributions of the underlying stochastic process.

Funding Statement

M. Engel has been supported by Germany’s Excellence Strategy—The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

Acknowledgments

We heartily thank M. Plum for several helpful discussions and references about rigorous eigenvalue bounds.

Citation

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Maxime Breden. Maximilian Engel. "Computer-assisted proof of shear-induced chaos in stochastically perturbed Hopf systems." Ann. Appl. Probab. 33 (2) 1252 - 1294, April 2023. https://doi.org/10.1214/22-AAP1841

Information

Received: 1 March 2021; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1519.34039
MathSciNet: MR4564426
Digital Object Identifier: 10.1214/22-AAP1841

Subjects:
Primary: 35P99 , 37-04 , 37M25
Secondary: 37H15 , 60J99

Keywords: homotopy method , Kolmogorov operators , Lyapunov exponents , quasi-ergodic distribution

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.33 • No. 2 • April 2023
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