Geodesic
On the Existence and Stability of an Infinite-Dimensional Invariant Torus
Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 508-528 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider an annular set of the form $K=B\times \mathbb{T}^{\infty}$, where $B$ is a closed ball of the Banach space $E$, $\mathbb{T}^{\infty}$ is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps $\Pi\colon K\to K$, we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form $$ A=\{(v, \varphi)\in K: v=h(\varphi)\in E,\,\varphi\in\mathbb{T}^{\infty}\}, $$ where $h(\varphi)$ is a continuous function of the argument $\varphi\in\mathbb{T}^{\infty}$. We also study the question of the $C^m$-smoothness of this manifold for any natural $m$.
Keywords: mapping, infinite-dimensional invariant torus, stability, smoothness.
Mots-clés : annulus principle 
@article{MZM_2021_109_4_a2,
     author = {S. D. Glyzin and A. Yu. Kolesov and N. Kh. Rozov},
     title = {On the {Existence} and {Stability} of an {Infinite-Dimensional} {Invariant} {Torus}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {508--528},
     year = {2021},
     volume = {109},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a2/}
}
TY  - JOUR
AU  - S. D. Glyzin
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
TI  - On the Existence and Stability of an Infinite-Dimensional Invariant Torus
JO  - Matematičeskie zametki
PY  - 2021
SP  - 508
EP  - 528
VL  - 109
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a2/
LA  - ru
ID  - MZM_2021_109_4_a2
ER  - 
%0 Journal Article
%A S. D. Glyzin
%A A. Yu. Kolesov
%A N. Kh. Rozov
%T On the Existence and Stability of an Infinite-Dimensional Invariant Torus
%J Matematičeskie zametki
%D 2021
%P 508-528
%V 109
%N 4
%U http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a2/
%G ru
%F MZM_2021_109_4_a2
S. D. Glyzin; A. Yu. Kolesov; N. Kh. Rozov. On the Existence and Stability of an Infinite-Dimensional Invariant Torus. Matematičeskie zametki, Tome 109 (2021) no. 4, pp. 508-528. http://geodesic.mathdoc.fr/item/MZM_2021_109_4_a2/

[1] L. P. Shil'nikov, D. V. Turaev, “Simple bifurcations leading to hyperbolic attractors”, Comput. Math. Appl., 34:2-4 (1997), 173–193 | MR | Zbl

[2] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Solenoidalnye attraktory diffeomorfizmov koltsevykh mnozhestv”, UMN, 75:2.(452) (2020), 3–60 | DOI | Zbl

[3] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. Chua, Metody kachestvennoi teorii v nelineinoi dinamike, Ch. 1, In-t kompyuternykh issledovanii, M.–Izhevsk, 2004 | MR

[4] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “O nekotorykh dostatochnykh usloviyakh giperbolichnosti”, Differentsialnye uravneniya i dinamicheskie sistemy, Tr. MIAN, 308, MIAN, M., 2020, 116–134 | DOI