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，其状态变量呈现正常或反常的扩散运动。对导致正常或反常扩散的机制有了透彻的理解，主要针对低维动力系统。最近，这种系统中的遍历破坏问题再次引发了人们的兴趣。另一方面，时滞系统是非常普遍和实际相关的系统，其中显式地考虑了反应的有限传输时间。这些形式上的无限维系统有很长的历史，但令人惊讶的是，这种系统中扩散运动的出现几乎没有被研究过。我们最近发现的层流混沌是时滞系统中一种新的混沌运动形式，它也为时滞系统中的正常扩散和异常扩散提供了迄今未知的机制，从而连接了非线性动力学的两个传统分支。这个项目的总体目标是探索这类时滞系统中扩散运动的性质，包括遍历破坏问题，以及从层流到湍流混沌过渡中出现的情景。

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.