Beltrami fields exhibit knots and chaos almost surely
In this paper, we show that, with probability $1$ , a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 s...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S205050942300052X/type/journal_article |
| _version_ | 1797794590391009280 |
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| author | Alberto Enciso Daniel Peralta-Salas Álvaro Romaniega |
| author_facet | Alberto Enciso Daniel Peralta-Salas Álvaro Romaniega |
| author_sort | Alberto Enciso |
| collection | DOAJ |
| description | In this paper, we show that, with probability
$1$
, a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including Kolmogorov–Arnold–Moser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on
${\mathbb {R}}^3$
and of high-frequency Beltrami fields on the 3-torus. |
| first_indexed | 2024-03-13T03:04:23Z |
| format | Article |
| id | doaj.art-d840b9dc2bd7463b8825c1f25d3ef825 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-13T03:04:23Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-d840b9dc2bd7463b8825c1f25d3ef8252023-06-27T08:12:30ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.52Beltrami fields exhibit knots and chaos almost surelyAlberto Enciso0https://orcid.org/0000-0002-9039-1863Daniel Peralta-Salas1https://orcid.org/0000-0001-5567-8538Álvaro Romaniega2https://orcid.org/0000-0003-4154-8681Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain; E-mail:Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain; E-mail:Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, 28049 Madrid, Spain; E-mail:In this paper, we show that, with probability $1$ , a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of stationary Euler flows in dimension 3, is V.I. Arnold’s 1965 speculation that a typical Beltrami field exhibits the same complexity as the restriction to an energy hypersurface of a generic Hamiltonian system with two degrees of freedom. The proof hinges on the obtention of asymptotic bounds for the number of horseshoes, zeros and knotted invariant tori and periodic trajectories that a Gaussian random Beltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodin theory for Gaussian random monochromatic waves and the application of different tools from the theory of dynamical systems, including Kolmogorov–Arnold–Moser (KAM) theory, Melnikov analysis and hyperbolicity. Our results hold both in the case of Beltrami fields on ${\mathbb {R}}^3$ and of high-frequency Beltrami fields on the 3-torus.https://www.cambridge.org/core/product/identifier/S205050942300052X/type/journal_article35Q3160G1537H05 |
| spellingShingle | Alberto Enciso Daniel Peralta-Salas Álvaro Romaniega Beltrami fields exhibit knots and chaos almost surely Forum of Mathematics, Sigma 35Q31 60G15 37H05 |
| title | Beltrami fields exhibit knots and chaos almost surely |
| title_full | Beltrami fields exhibit knots and chaos almost surely |
| title_fullStr | Beltrami fields exhibit knots and chaos almost surely |
| title_full_unstemmed | Beltrami fields exhibit knots and chaos almost surely |
| title_short | Beltrami fields exhibit knots and chaos almost surely |
| title_sort | beltrami fields exhibit knots and chaos almost surely |
| topic | 35Q31 60G15 37H05 |
| url | https://www.cambridge.org/core/product/identifier/S205050942300052X/type/journal_article |
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