Parametrised KAM Theory, an Overview
Author(s):
Broer H. W., Hanßmann H., Wagener F. O.
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
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