ABSTRACT

Kolmogorov-Arnol'd-Moser Theory classically was mainly developed for conservative systems, establishing persistence results for quasi-periodic invariant tori in nearly integrable systems. In this survey we focus on dissipative systems, where similar results hold. In non-conservative settings often parameters are needed for the persistence of invariant tori. When considering families of such dynamical systems bifurcations of quasi-periodic tori may occur. As an example we discuss the quasi-periodic Hopf bifurcation.

Introduction

Motivation

Kolmogorov-Arnold-Moser Theory is concerned with the occurrence of multior quasi-periodic invariant tori in nearly integrable systems. Integrable systems by definition have a toroidal symmetry which produces invariant tori as orbits under the corresponding torus action. The central problem of KAM Theory, is the continuation of such tori to nearly integrable perturbations of the system.

Initially this part of perturbation theory was developed for conservative, i.e., Hamiltonian, systems that model the frictionless dynamics of classical mechanics. Related physical questions are concerned with the perpetual stability of the solar system, of tokamak accelerators, etc. Initiated by Poincaré at the end of the 19th century, the theory was further developed by Birkhoff and Siegel and later established by Kolmogorov, Arnold, Moser and others from the 1950s on. For a historical overview and further reference, see. As pointed out in and later in, the conservative approach can be extended to many other settings, like to general dissipative systems, to volume preserving systems and to various equivariant or reversible settings.