Aspekte niedrig-dimensionaler und nicht-autonomer Dynamischer Systeme

遍历理论和动力系统已经在基本上所有的科学和许多其他数学领域找到了它们的应用。这个研究项目的目的是通过分析几种类型的低维和/或非自治动力系统来为这一领域做出实质性的贡献。正在审议的系统的一个共同特征是缺乏或不相关的周期轨道。这使得他们的研究特别困难，因为在绝大多数情况下，动力系统的长期行为是众所周知的，周期轨道的存在是一个关键特征，而且往往是任何分析的起点。与此形成鲜明对比的是，人们对周期无点动力学的理解仍然很少，因此需要新的概念和思想来描述它们。特别需要解决的问题包括所谓的奇异非混沌吸引子在拟周期强迫圆映射分类中的作用，非自治分叉理论，特别关注非自治Hopf分叉的两步情景，以及同胚动力学的分类。在物理学和生物学中的应用，应该同样考虑，包括描述多频率受迫振荡器和在外部噪声影响下的神经元编码。

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.