On the Role of Frustration in Couplex Dynamical Systems

在物理学的背景下，自旋系统引入了受挫的概念。在这个项目中，我们想要将挫折的概念扩展到不是由作用量或哈密顿量衍生的动力系统，并研究挫折在其各种实现中的作用。这类系统是一般振子、可激发介质和耦合映射。我们将研究受挫单位作为分层组织系统中双稳态和多稳态系统的基石的作用。与自旋系统类似，我们预计吸引子的相空间将在定量和定性方面得到丰富，更长的驰豫时间将作为玻璃行为的残余物达到稳定状态。不同的吸引子可能与不同的功能有关。功能良好的生物系统如果同时是稳定和灵活的，它们就能有效地运行。我们的猜测是，适当的中等程度的挫折感可以实现这一目标。虽然高度的挫折感使系统对扰动不稳定，但过低的挫折度使系统变得僵硬，容易受到功能单元的移除。关于灵活性方面，我们进一步期望，在层级组织的每一级上设计具有一定数量的受挫单元的系统是确保丰富的相空间和高性能灵活性的有效方式，而不需要在单个级别上使用复杂的单元。博士论文的项目应该会产生验证这些猜想的结果。虽然第一部分将讨论各种动力系统中的挫折感的一般概念，但第二部分将重点关注由耦合阻遏和激活子组成的空间组织的遗传系统，并详细研究它们的相空间以及它们对于不同类型扰动的稳定性。

In the context of physics the notion of frustration was introduced for spin systems. In this project we want to extend the concept of frustration to dynamical systems that do not derive from an action or Hamiltonian and study frustration in its various realizations. Such systems are generic oscillators, excitable media, and coupled maps. We shall study the role of frustrated units as building blocks in bi-and multistable systems in hierarchically organized systems. In analogy to spin systems, we expect an enrichment of the phase space of attractors, both in quantitative and qualitative terms, and longer relaxation times to reach a stationary state as remnant of glass-like behavior. Different attractors may be related to different functions. Well functioning biological systems perform efficiently if they are stable and flexible at the same time. Our conjecture is that an appropriate intermediate degree of frustration can achieve this goal. While a high degree of frustration makes the system unstable against perturbations, a too low degree makes it stiff and vulnerable against removal of functional units. Related to the aspect of flexibility we further expect that designing systems with a certain number of frustrated units on each level of a hierarchical organization is an efficient way of ensuring a rich phase space and a high flexibility in performance, without the need for complex units on the individual level. The project of the PhD thesis should yield results towards verification of these conjectures. While the first part will deal with the generic concept of frustration in various dynamical systems, the second part shall focus on spatially organized, genetic systems of coupled repressors and activators and study their phase space as well as their stability properties with respect to different type of perturbations in detail.