Annulus principle in the problem of the existence of an infinite-dimensional invariant torus

S. D. Glyzin, A. Yu. Kolesov

P.G. Demidov Yaroslavl State University

Abstract: An annulus-like set of the form $K=B\times\mathbb{T}^{\infty}$ is under consideration, where $B$ is a closed ball in a Banach space $V$ and $\mathbb{T}^{\infty}$ is the so-called standard infinite-dimensional torus, defined by $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional Banach space and $\mathbb{Z}^{\infty}$ is an abstract integer lattice in $E$. The main result is as follows: 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 $A=\{(v,\varphi)\in K\colon v=h(\varphi)\in V,\break \varphi\in\mathbb{T}^{\infty}\}$, where $h(\varphi)$ is a continuous function of $\varphi\in\mathbb{T}^{\infty}$. We also answer a number of related questions. First, we consider the problem of the $C^m$-smoothness of the manifold $A$ for each positive integer $m$; second, we show that all trajectories of the map $\Pi$ with initial conditions in $K$ tend to $A$ while admitting an asymptotic phase; third, we extend our results to semiflows and then apply the theory developed to integral networks of nonlinear oscillators.

Keywords: map, annulus principle, infinite-dimensional invariant torus, stability, smoothness, semiflow, integral networks.

UDC: 517.926

MSC: 37D20, 46T20

Received: 18.11.2025

DOI: 10.4213/rm10291