Fast-slow dynamical systems

在自然界中，复杂系统的行为取决于不同元素的相互作用，这些元素在空间和时间上以不同的尺度进化。例如，在统计力学中，人们可以很容易地用宏观相互作用的时间标度来区分分子相互作用的时间标度。在天体力学中，不同类型的运动根据不同的时间尺度而演变。在生物学中，细胞的寿命可能比它所构成的有机体的寿命短几个数量级；同样，单个个体的生死对其所属物种的进化影响不大。******然而，当一个人想把这样的系统作为一个动力系统来建模时，由于几个时间尺度的存在而产生的复杂性产生了巨大的困难，这些困难长期以来阻碍了对这些系统动力学的完整和全面的理解。******多尺度系统最简单的可能情况是只考虑两个不同的尺度。研究这类系统的一个非常巧妙的想法是，假设快速系统进化得如此之快，以至于在缓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，这在某些自然假设下是可以很好地理解的。这个思想已经在所谓的平均理论中实现了混沌动力学的设置。******在我提出的研究计划中，根据我和我的合著者在过去几年中获得的平均理论中最近的一些重要结果，我想深入探索这样的系统。这将使我们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.