Aspekte niedrig-dimensionaler und nicht-autonomer Dynamischer Systeme

低维和非自治动态系统的方面

基本信息

批准号：
129048169
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：
Independent Junior Research Groups
财政年份：
2009
资助国家：
德国
起止时间：
2008-12-31 至 2015-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/129048169?language=en
关键词：
Aspekte niedrig dimensionaler und nicht

项目摘要

Ergodic Theory and Dynamical Systems have found their applications in basically all sciences and many other fields of mathematics. The aim of this research project is to make substantial contributions to this field, by analysing several types of low-dimensional and/or non-autonomous dynamical systems. A common feature of the systems under consideration is the lack, or irrelevance, of periodic orbits. This renders their investigation particularly difficult, since in the vast majority of situations where the long-term behaviour of a dynamical system is well-understood, the existence of periodic orbits is a crucial feature and often the starting point of any analysis. In contrast to this, periodic point free dynamics are still rather poorly understood in general, such that new concepts and ideas are required for their description. The issues which should be addressed in particular include the role of so-called strange nonchaotic attractors in the classification of quasiperiodically forced circle maps, nonautonomous bifurcation theory with a particular focus on the two-step scenario for the non-autonomous Hopf bifurcation, and the classification of the dynamics of toral homeomorphisms. Applications to physics and biology, which shall be considered likewise, include the description of multi-frequency forced oscillators and neuronal coding under the influence of external noise.

千古理论和动力学系统在基本上在所有科学和许多其他数学领域中都发现了它们的应用。该研究项目的目的是通过分析几种类型的低维和/或非自主动力学系统来为该领域做出重大贡献。所考虑的系统的一个共同特征是周期性轨道的缺乏或无关。这使他们的调查特别困难，因为在绝大多数情况下，动态系统的长期行为是充分理解的，周期性轨道的存在是至关重要的特征，通常是任何分析的起点。与此相反，一般而言，周期性的自由动力学仍然对其描述需要新的概念和想法。应特别解决的问题包括所谓的奇怪的非偶然吸引子在Quasiperiodododed迫切圆形地图的分类中的作用，非自治分叉理论，特别关注非自治的Hopf分歧的两步场景，以及Toral Homanomormorphismssss symics的动态分类。在外部噪声的影响下，应同样考虑到物理和生物学的应用包括对多频振荡器和神经元编码的描述。