Chaotic Diffusion in Delay Systems

混沌扩散是非线性动力系统中一种众所周知的确定性现象，其状态变量表现为正常或反常的扩散运动。深入了解导致正常或异常扩散的机制主要是在低维动力系统中实现的。最近，这种系统的遍历性断裂问题引发了人们的新兴趣。另一方面，延迟系统是一种非常普遍和实际相关的系统，它明确地考虑了反应的有限传输时间。这些正式的无限维系统有着悠久的历史，但令人惊讶的是，扩散运动在这些系统中的出现几乎没有被研究过。我们最近发现的层流混沌是延迟系统中混沌运动的一种新形式，也为延迟系统中的正常和异常扩散提供了迄今为止未知的机制，从而连接了非线性动力学的两个传统分支。本项目的总体目标是探索这种延迟系统的扩散运动特性，包括遍历性破缺问题，以及从层流到湍流的过渡中出现的场景。

Chaotic diffusion is a well-known deterministic phenomenon in nonlinear dynamical systems, where the state variable shows normal or anomalous diffusive motion. A thorough understanding of the mechanisms leading to normal or anomalous diffusion was achieved mainly for low-dimensional dynamical systems. Recently renewed interest was triggered by questions of ergodicity breaking in such systems. Delay systems, on the other hand, are very general and practically relevant systems, where finite transmission times of reactions are explicitly taken into account. These formally infinite-dimensional systems have a long history, but surprisingly, the appearance of diffusive motion in such systems has hardly been investigated. Our recent detection of laminar chaos, a new form of chaotic motion in delay systems, provides also hitherto unknown mechanisms for normal and anomalous diffusion in delay systems, thus connecting two traditional branches of nonlinear dynamics. The general goal of this project is to explore the properties of diffusive motion in such delay systems, including questions of ergodicity breaking, and the scenarios appearing in the transition from laminar to turbulent chaos.