Dynamical behaviour and ergodic theory of quasiperiodically forced maps, with particular attention to the existence and properties of strange non-chaotic attractors

• Applicant: Professor Dr. Tobias Henrik Oertel-Jäger
• Subject Area: Mathematics
• Term: from 2006 to 2009
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 25618165

Many dynamical systems both of practical and theoretical importance are subject to external forcing. While the influence of periodic forcing is quite wellunderstood, mathematical results on quasiperiodically forced (qpf) systems are still rather few. The subject of the proposed research project is therefore a systematic study of the dynamical properties and long-time behavior of one of the most important classes of qpf systems, namely qpf circle homeo- and diffeomorphisms. Thereby, the main focus will be two-fold: The first aim is to obtain classification results. First steps in this direction already indicate that such a classification might be in partial analogy to the respective results from one-dimensional (unforced) dynamics, but there is a richer variety of possible behavior and new interesting phenomena show up which do not occur in the one-dimensional setting. Ultimately, such studies should also provide the basis for the understanding of phenomena like mode-locking and the structure of Arnold tongues in parameterized families such as the qpf Arnold circle map. The second objective is the investigation of so-called strange non-chaotic attractors, which seem to occur frequently in quasiperiodically forced systems. Such attractors exhibit the very unusual combination of a strange geometrical and topological structure with non-chaotic dynamics. Consequently these objects have evoked considerable interest in theoretical physics, but so far rigorous results are still few and a proof of their existence is available only in a few very particular situations. However, two recent approaches to this problem seemingly allow for much greater generalization. This would mean that for the first time the widespread existence of SNA in a variety of different models could be proved rigorously. Finally, a third objective of the applicant is to acquire a deeper knowledge about related systems in two and three-dimensional dynamics, in order to diversify his research interests in this direction. As an example, some aspects from the theory of general torus horneomorphisms and irrational pseudo-translations of the torus are included, which might provide a starting point for further research.

• DFG Programme: Research Fellowships
• International Connection: France
• Host: Professor Jean-François Rigoni