Invariant tori via higher order averaging method: existence, regularity, convergence, stability, and dynamics

• Published in: Mathematische annalen Vol. 389; no. 1; pp. 543 - 590
• Main Authors: Novaes, Douglas D., Pereira, Pedro C. C. R.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-05-2024; Springer Nature B.V
• Subjects: Asymptotic properties; Convergence; Dynamic stability; Dynamic structural analysis; Invariants; Manifolds; Mathematics; Mathematics and Statistics; Orbits; Regularity; Toruses; 34C23; 34C45; 34C29
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-023-02654-2

Important information about the dynamical structure of a differential system can be revealed by looking into its invariant compact manifolds, such as equilibria, periodic orbits, and invariant tori. This knowledge is significantly increased if asymptotic properties of the trajectories nearby such invariant manifolds can be determined. In this paper, we present a result providing sufficient conditions for the existence of invariant tori in perturbative differential systems. The regularity, convergence, and stability of such tori as well as the dynamics defined on them are also investigated. The conditions are given in terms of their so-called higher order averaged equations. This result is an extension to a wider class of differential systems of theorems due to Krylov, Bogoliubov, Mitropolsky, and Hale.