Generalizations of Laminar Chaos

该项目的目标是继续探索层流混沌概念的推广，我们发现层流混沌概念适用于具有周期性变化时滞的系统。我们希望沿着原计划的路线继续我们的研究，但也要增加以前没有考虑过的新的、有趣的方面。第一行是考虑具有越来越复杂的延迟参数的标量延迟微分方程：周期-准周期-随机-状态相关。这里还有待完成我们对随机延迟的研究，并考虑最复杂的状态相关延迟情况。第二行是关于非标量系统的推广，由于时间有限，我们还不能考虑。他们现在应该得到更多的关注，因为其他小组受到我们工作的启发，已经开始考虑在矢量系统的一个组成部分中显示层流混沌的影响，特别是同步的影响。然而，这方面的主要工作计划遵循我们从第一个资助期开始的原始建议。在发表我们的结果时，一个迄今为止被忽视的方面引起了我们的注意：我们以前所有的结果都是针对一个离散的时间相关延迟获得的。然而，在现实中，延迟时间可能受到小而快速的波动的扰动，这种波动可以用分布宽度小但非零的分布延迟来建模。一方面，人们会天真地期望层流混沌的概念在如此小的扰动中幸存下来，但另一方面，我们的基本工具之一，访问图，不再适用。对于广义延迟分布，无论是离散的还是连续的，我们都不期望层流混沌在一般情况下存在。但这就提出了一个问题，即时间相关的分布式延迟以何种方式允许或破坏层流混沌现象，这构成了一个新的、重要的第三条研究方向。相应的，计划工作可分为三个部分：1 .层流混沌对随机延迟（完备性和弱无序）系统和状态依赖延迟系统的推广。具有分布时滞的标量系统的层流混沌[j]。层流混沌概念在非标量系统中的推广。这三个部分在逻辑上是有联系的。I.和II.两者都处理标量系统，但在延迟参数中变得越来越复杂：然而理解随机延迟的层流混沌的发生是理解其在状态相关延迟系统中发生的先决条件，这些研究对于理解II中具有分布延迟的标量系统也是必要的。所有这些研究都旨在从几何角度理解抽象状态空间中导致层流混沌的定性动力学。这可能使我们最终不需要参考延迟方程的特定形式就可以定义它。

The goal of this project continues to be the exploration of the generalizations of the concept of Laminar Chaos, which we found for systems with periodically varying time-delay. We want to continue our research along the originally planned lines, but also add new, interesting aspects not considered previously. The first line was to consider scalar delay differential equations with increasing complexity in the delayed argument: periodic - quasi-periodic - random - state-dependent. Here it remains to finalize our studies on random delays and to consider the most complicated case of state-dependent delay. The second line was about generalizations to non-scalar systems, which we could not yet consider due to lack of time. They should now receive enhanced attention, because other groups, inspired by our work, started already to consider effects, especially of synchronization, for vectorial systems showing Laminar Chaos in one of its components. The main work in this line, however, is planned to follow our original proposal from the first funding period. A hitherto neglected aspect attracted our attention while publishing our results: all our previous results were obtained for one discrete time-dependent delay. In reality, however, the delay time may be perturbed by small, but fast fluctuations, which can be modelled by a distributed delay with a small, but non-zero width of its distribution. On one hand, one would naively expect that the concept of Laminar Chaos survives such small perturbations, but on the other hand, one of our fundamental tools, the access map, can no longer be applied. For broad delay distributions, discrete or continuous, we do not expect that Laminar Chaos survives in general. But this poses the question in which way time-dependent distributed delays allow or destroy the phenomenon of Laminar Chaos, which constitutes a new and important third line of research. Correspondingly, the planned work can be divided into three parts: I. Generalization of Laminar Chaos to systems with random delays (completion and weak disorder) and systems with state-dependent delays II. Laminar Chaos for scalar systems with distributed delays III. Generalization of the concept of Laminar Chaos to non-scalar systems. The three parts are logically connected. I. and II. both treat scalar systems, but with increasing complexity in the delayed argument: Whereas understanding the occurrence of Laminar Chaos for random delays is a prerequisite for understanding its occurrence in systems with state-dependent delay, these investigations are also necessary for understanding scalar systems with distributed delay in II. All these investigations also aim at a geometrical understanding of the qualitative dynamics in abstract state space leading to Laminar Chaos. This may allow us to define it eventually without reference to specific forms of the delay equations.