Three Examples in the Dynamical Systems Theory
We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms , of a closed two-dimensional annulus that possess the intersection property, but their composition does not ( being just the rotation by π/2). The seco...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2022
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211820 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Three Examples in the Dynamical Systems Theory. Mikhail B. Sevryuk. SIGMA 18 (2022), 084, 13 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862732807089946624 |
|---|---|
| author | Sevryuk, Mikhail B. |
| author_facet | Sevryuk, Mikhail B. |
| citation_txt | Three Examples in the Dynamical Systems Theory. Mikhail B. Sevryuk. SIGMA 18 (2022), 084, 13 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms , of a closed two-dimensional annulus that possess the intersection property, but their composition does not ( being just the rotation by π/2). The second example is that of a non-Lagrangian -torus ₀ in the cotangent bundle *ⁿ of ⁿ ( ≥ 2) such that ₀ intersects neither its images under almost all the rotations of *ⁿ nor the zero section of *ⁿ. The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form ẋ = (, ), ẏ = (, ) in the closed upper half-plane { ≥ 0} such that for each family, the corresponding phase portraits for 0 < < 1 and for > 1 are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.
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| first_indexed | 2026-04-17T15:40:36Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211820 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T15:40:36Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sevryuk, Mikhail B. 2026-01-12T10:19:07Z 2022 Three Examples in the Dynamical Systems Theory. Mikhail B. Sevryuk. SIGMA 18 (2022), 084, 13 pages 1815-0659 2020 Mathematics Subject Classification: 57R17; 53D12; 37C15 arXiv:2209.02620 https://nasplib.isofts.kiev.ua/handle/123456789/211820 https://doi.org/10.3842/SIGMA.2022.084 We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms , of a closed two-dimensional annulus that possess the intersection property, but their composition does not ( being just the rotation by π/2). The second example is that of a non-Lagrangian -torus ₀ in the cotangent bundle *ⁿ of ⁿ ( ≥ 2) such that ₀ intersects neither its images under almost all the rotations of *ⁿ nor the zero section of *ⁿ. The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form ẋ = (, ), ẏ = (, ) in the closed upper half-plane { ≥ 0} such that for each family, the corresponding phase portraits for 0 < < 1 and for > 1 are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology. I am grateful to L.V. Polterovich for interesting discussions of some aspects of symplectic topology and several useful references (in particular, the preprint [31]). Special thanks also to V.M. Zyuzkov. I am appreciative for helpful critical remarks of anonymous referees and of the editor’s comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Three Examples in the Dynamical Systems Theory Article published earlier |
| spellingShingle | Three Examples in the Dynamical Systems Theory Sevryuk, Mikhail B. |
| title | Three Examples in the Dynamical Systems Theory |
| title_full | Three Examples in the Dynamical Systems Theory |
| title_fullStr | Three Examples in the Dynamical Systems Theory |
| title_full_unstemmed | Three Examples in the Dynamical Systems Theory |
| title_short | Three Examples in the Dynamical Systems Theory |
| title_sort | three examples in the dynamical systems theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211820 |
| work_keys_str_mv | AT sevryukmikhailb threeexamplesinthedynamicalsystemstheory |