Abstract:
We consider an annular set of the form K=B×T∞, where B is a closed ball of the Banach space E, T∞ 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 Π:K→K, we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form A={(v,φ)∈K:v=h(φ)∈E,φ∈T∞}, where h(φ) is a continuous function of the argument φ∈T∞. We also study the question of the Cm-smoothness of this manifold for any natural m.
Citation:
S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “On the Existence and Stability of an Infinite-Dimensional Invariant Torus”, Mat. Zametki, 109:4 (2021), 508–528; Math. Notes, 109:4 (2021), 534–550