A superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n < d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d − n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as Cantor families of d-tori. A perturbation of a superintegrable system that admits a better approximation by a non-degenerate integrable perturbation of the superintegrable system is said to remove the degeneracy. In the minimal case d = n + 1 this can be achieved by means of averaging, but the more integrals of motion the superintegrable system admits the more difficult becomes the perturbation analysis.

This issue of Discrete and Continuous Dynamical Systems, Series S, is a collection of papers in the area of KAM theory and its applications.KAM theory, named after its founders A.N. Kolmogorov [1, 2], V.I. Arnol ′ d [3, 4, 5], and J.K. Moser [6,7], is a major part of Dynamical Systems Theory and Qualitative Theory of Differential Equations, both ordinary and partial (the descriptive term "KAM theory" was coined in a 1968 Russian preprint by F.M. Izrailev and B.V. Chirikov). It studies the typical occurrence of quasi-periodicity in nonintegrable dynamical systems and is often regarded as one of the foremost branches of modern Nonlinear Dynamics.The prototype of almost all the results in KAM theory is Kolmogorov's famous theorem [1] asserting that under small Hamiltonian perturbations of a completely integrable Hamiltonian system with n degrees of freedom, the unperturbed invariant n-tori carrying quasi-periodic motions with Diophantine frequencies are not destroyed but only slightly deformed in the phase space-provided that the unperturbed system satisfies a certain nondegeneracy (or transversality) condition. These perturbed tori are called KAM tori.KAM theory has had a great impact on the development of Mechanics and Physics; for instance, it refutes the Ergodic and Quasi-Ergodic Hypotheses in Statistical Hamiltonian Mechanics. Most of the applications of KAM theory concern stability problems, like the stability of the Solar System in Celestial Mechanics or perpetual conservation of adiabatic invariants [5]. There are also applications that relate to, e.g., calculating the short-wave approximations for the eigenvalues and eigenfunctions of various operators in Quantum Mechanics.Although the mainstream of KAM theory is in Hamiltonian systems, it became clear early, mainly from [7], that this approach similarly allows to explore invariant tori of all the possible dimensions, carrying quasi-periodic motions, in other conservative settings as well, e.g. in volume-preserving or reversible systems. In dissipative systems, one usually needs external parameters to make quasi-periodic dynamics occur in a robust way. The families of quasi-periodic attractors typically met in the dissipative setting can figure as a transient stage to chaos. Similarly Aubry-Mather sets manifest themselves as a transient stage in the conservative setup. Besides looking for invariant tori in dynamical systems, KAM theory also includes other problems where small divisors are encountered, e.g. normalizations of vector fields and mappings [3].Most of the recent achievements in KAM theory concentrate around quasiperiodicity in infinite dimensional systems (partial differential equations), quasiperiodic bifurcations describing transitions between qualitatively different kinds of dynamics, the persistence of invariant tori under weak nondegeneracy conditions (with only partial preservation of frequencies), the Gevrey-smoothness of Cantor families of invariant tori in Gevrey-smooth or analytic systems, and quasiperiodicity in generalized Hamiltonian...

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