Generalizations of Laminar Chaos

层流混沌的概括

• 批准号: 456546951
• 负责人: Professor Dr. Günter Radons
• 金额: --
• 依托单位: Forschungsgruppe Komplexe Systeme und Nichtlineare Dynamik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/456546951?language=en
• 关键词: Generalizations; Laminar; Chaos

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.

这个项目的目标仍然是探索层流混沌概念的推广，我们发现系统具有周期性变化的时间延迟。我们希望继续我们的研究沿着最初计划的路线，但也增加了新的，有趣的方面没有考虑以前。第一行是考虑标量延迟微分方程的延迟参数的复杂性增加：周期-拟周期-随机-状态依赖。在这里，它仍然是完成我们的研究随机延迟和考虑最复杂的情况下，状态依赖延迟。第二行是关于非标量系统的推广，由于时间不够，我们还不能考虑。他们现在应该得到更多的关注，因为其他团体，受到我们的工作的启发，已经开始考虑的影响，特别是同步，矢量系统显示层流混沌在其组成部分之一。然而，这一领域的主要工作计划遵循我们从第一个供资期开始的最初建议。一个迄今为止被忽视的方面吸引了我们的注意，而发表我们的结果：我们以前的所有结果得到一个离散的时间依赖延迟。然而，实际上，延迟时间可能会受到小但快速波动的干扰，这可以通过具有小但非零分布宽度的分布式延迟来建模。一方面，人们会天真地认为层流混沌的概念能够经受住如此小的扰动，但另一方面，我们的基本工具之一，访问映射，不再适用。对于宽延迟分布，离散或连续，我们不希望层流混沌生存一般。但这就提出了一个问题，即时间相关的分布延迟是以何种方式允许或破坏层流混沌现象的，这构成了一个新的和重要的第三条研究路线。相应地，计划的工作可以分为三个部分：一。将层流混沌推广到随机时滞系统（完备和弱无序）和状态依赖时滞系统2。具有分布时滞标量系统的层流混沌III。将层流混沌概念推广到非标量系统。这三个部分在逻辑上是相互联系的。I.和二.都处理标量系统，但在延迟的论点增加了复杂性：虽然理解随机延迟的层流混沌的发生是理解其发生在状态依赖延迟的系统的先决条件，这些调查也是理解标量系统与分布延迟在II。所有这些调查的目的也是在抽象的状态空间中的定性动力学的几何理解导致层流混沌。这使得我们最终可以定义它，而不需要参考延迟方程的具体形式。