Fast-slow dynamical systems

在自然界中，复杂系统是根据不同元素的相互作用而表现出来的，这些元素在空间和时间上以不同的尺度演化。例如：在统计力学中，人们可以很容易地通过宏观相互作用的时间尺度来分离分子相互作用的时间尺度。在天体力学中，不同类型的运动根据截然不同的时间尺度演化。在生物学中，细胞的寿命可能比它构成的有机体的寿命短几个数量级；同样，单个个体的生死对该个体所属物种的进化几乎没有影响。然而，当人们想要将这样的系统建模为动态系统时，由于存在多个时间尺度而产生的复杂性产生了巨大的困难，这在很长一段时间内阻碍了对这种系统动力学的全面和全面的了解。对于多尺度系统，最简单的可能情况是只有两个不同的尺度可供考虑。研究这类系统的一个非常巧妙的想法是，假设快系统进化得如此之快，以至于在慢系统有机会进化之前，它达到了某种平衡。因此，慢系统将受到快系统的平均行为的影响，在一些自然假设下，这是可以相当好地理解的。这个想法已经在所谓的平均理论中的混沌动力学的背景下得到了实现。在我提出的研究计划中，我和我的合著者在过去几年中获得了一些关于平均理论的最新重要成果，我想要对这类系统进行非常深入的探索。这将使我们能够在很大程度上理解混沌性质是如何在这个框架中产生和表现的。

It is commonplace in nature that complex systems behave according to the interaction of different elements, which evolve with different scales in space and time. For instance: in statistical mechanics one can easily separate the timescale of molecular interaction by the timescale of macroscopic interaction. In celestial mechanics, different type of motions evolve according to dramatically different timescales. In biology, the lifespan of a cell might be several order of magnitude smaller than the lifespan of the organism it constitutes; and, similarly, life or death of a single individual has little impact on the evolution of the species that that individual belongs to.******Yet, when one wants to model such systems as a dynamical system, the complexity arising by the presence of several timescales generates enormously difficult aspects which have for a long time prevented a complete and comprehensive understanding of the dynamics of such systems.******The simplest possible case of a multi-scale system is when there are only two different scales to be considered. A very ingenious idea to study such systems is to assume the fast system to evolve so rapidly that it reaches some sorts of equilibrium before the slow system has a chance to evolve. The slow system will therefore be affected by the averaged behavior of the fast system, which under some natural assumptions can be fairly well understood. This idea has been implemented in the setting of chaotic dynamics in the so-called Averaging Theory.******In my proposed research program, building from some recent important results in Averaging Theory that my coauthors and I have obtained in the last couple of years, I would like to explore such systems to a very deep level. This would allow to understand to a great extent how chaotic properties arise and behave in this framework.