Parametrised KAM Theory, an Overview

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.