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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.

遍历理论和动力系统在基本上所有的科学和许多其他数学领域都有应用。该研究项目的目的是通过分析几种类型的低维和/或非自治动力系统，为这一领域做出实质性贡献。所考虑的系统的一个共同特点是缺乏或不相关的周期轨道。这使得他们的调查特别困难，因为在绝大多数情况下，长期行为的动力系统是很好的理解，周期轨道的存在是一个至关重要的特征，往往是任何分析的起点。与此相反，周期性点自由动力学仍然是相当少的理解，在一般情况下，这样的新概念和想法是必要的，他们的描述。特别是应解决的问题包括所谓的奇怪的非混沌吸引子的作用，在准周期强迫圆映射的分类，非自治分支理论，特别侧重于两个步骤的情况下，非自治的Hopf分支，和分类的动态的toral同胚。在物理学和生物学上的应用，也应该同样考虑，包括描述多频受迫振荡器和在外部噪声影响下的神经元编码。