Parametrised KAM Theory, an Overview

Kolmogorov – Arnold – Moser theory started in the 1950s as theperturbation theory for persistence of multi- orquasi-periodic motions in Hamiltonian systems.Since then the theory obtained a branch where the persistentoccurrence of quasi-periodicity is studied in variousclasses 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 setof positive Hausdorff measure in a sub-manifold of theproduct of phase space and parameter space.This paper contains an overview of this development withan emphasis on the world of dissipative systems, wherefamilies of quasi-periodic tori occur and bifurcate in apersistent way.The transition from orderly to chaotic dynamics here formsa leading thought.