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

准周期强迫映射的动力学行为和遍历理论，特别关注奇怪的非混沌吸引子的存在和性质

• 批准号: 25618165
• 负责人: Professor Dr. Tobias Henrik Oertel-Jäger
• 金额: --
• 依托单位: Lehrstuhl für Ergodentheorie und Dynamische Systeme
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2006
• 资助国家: 德国
• 起止时间: 2005-12-31 至 2008-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/25618165?language=en
• 关键词: Dynamical; behaviour; ergodic; theory; quasiperiodically

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.

许多具有实际意义和理论意义的动力系统都受到外力的影响。虽然周期强迫的影响已经被很好地理解，但关于准周期强迫系统的数学结果仍然相当少。因此，该研究项目的主题是系统地研究qpf系统中最重要的一类系统的动力学性质和长期行为，即qpf圆同胚和微分同胚。因此，主要的重点将是双重的：第一个目标是获得分类结果。在这个方向上的第一步已经表明，这种分类可能部分类比于一维（非强制）动力学的各自结果，但是有更丰富的各种可能的行为和新的有趣的现象出现，这些现象在一维设置中不会发生。最终，这些研究还应该为理解锁模等现象以及参数化族（如qpf Arnold圆图）中Arnold舌的结构提供基础。第二个目标是研究所谓的奇异非混沌吸引子，它似乎经常出现在准周期强迫系统中。这种吸引子表现出一种奇怪的几何和拓扑结构与非混沌动力学的非常不寻常的组合。因此，这些物体在理论物理学中引起了相当大的兴趣，但到目前为止，严格的结果仍然很少，而且它们的存在只能在少数非常特殊的情况下得到证明。然而，最近有两种解决这个问题的方法似乎可以进行更广泛的推广。这将意味着可以首次严格证明SNA在各种不同模型中的广泛存在。最后，申请人的第三个目标是获得二维和三维动力学相关系统的更深层次的知识，以便在这个方向上多样化他的研究兴趣。以环面一般同胚理论和环面不合理伪平移理论为例，为进一步的研究提供了一个起点。