This article is cited in 1 paper

Parametrised KAM Theory, an Overview

Henk W. Broer^a, Heinz Hanßmann^b, Florian Wagener<sup>c

a</sup> Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Rijksuniversiteit Groningen, Postbus 407, 9700 AK Groningen, The Netherlands
^b Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, 3508 TA Utrecht, The Netherlands
^c Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Universiteit van Amsterdam, Postbus 15867, 1001 NJ Amsterdam, The Netherlands

Abstract: Kolmogorov – Arnold – Moser theory started in the 1950s as the perturbation theory for persistence of multi- or quasi-periodic motions in Hamiltonian systems. Since then the theory obtained a branch where the persistent occurrence of quasi-periodicity is studied in various classes of systems, which may depend on parameters. The view changed into the direction of structural stability, concerning the occurrence of quasi-periodic tori on a set of positive Hausdorff measure in a sub-manifold of the product of phase space and parameter space. This paper contains an overview of this development with an emphasis on the world of dissipative systems, where families of quasi-periodic tori occur and bifurcate in a persistent way. The transition from orderly to chaotic dynamics here forms a leading thought.

Keywords: quasi-periodic invariant tori, KAM theory, persistence, bifurcations

MSC: 37C55, 70K43, 34C23

Received: 24.05.2024
Accepted: 12.02.2025

Language: English

DOI: 10.1134/S156035472551001X