On the Existence and Stability of an Infinite-Dimensional Invariant Torus
| Autor: | N. Kh. Rozov, Sergey Dmitrievich Glyzin, A. Yu. Kolesov |
|---|---|
| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Mathematical Notes. 109:534-550 |
| ISSN: | 1573-8876 0001-4346 |
| DOI: | 10.1134/s0001434621030226 |
| Popis: | 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 $$m$$ . |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|