Florian Wagener

Near-resonances between frequencies notoriously lead to small denominators when trying to prove persistence of invariant tori carrying quasi-periodic motion. In dissipative systems external parameters detuning the frequencies are needed so that Diophantine conditions can be formulated, which allow to solve the homological equation that yields a conjugacy between perturbed and unperturbed quasi-periodic tori. The parameter values for which the Diophantine conditions are not fulfilled form so-called resonance gaps. Normal hyperbolicity can guarantee invariance of the perturbed tori, if not their quasi-periodicity, for larger parameter ranges. For a 1-dimensional parameter space this allows to close almost all resonance gaps.

In the skew Hopf bifurcation a quasi-periodic attractor with nontrivial normal linear dynamics loses hyperbolicity. Periodic, quasi-periodic and chaotic dynamics occur, including motion with mixed spectrum. The case of $3$-dimensional skew Hopf bifurcation families of diffeomorphisms near integrability is discussed, surveying some recent results in a broad perspective. One result, using KAM-theory, deals with the persistence of quasi-periodic circles. Other results concern the bifurcations of periodic attractors in the case of resonance.

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.