Fast-slow dynamical systems

快-慢动力系统

基本信息

批准号：
RGPIN-2017-06619
负责人：
DeSimoi, Jacopo
金额：
$ 1.82万
依托单位：
University of Toronto
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2020
资助国家：
加拿大
起止时间：
2020-01-01 至 2021-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712281
关键词：
Fast slow dynamical systems

项目摘要

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.

在自然界中，复杂系统根据不同元素的相互作用而表现，这些元素在空间和时间上以不同的尺度演化，这是很常见的。例如：在统计力学中，我们可以很容易地将分子相互作用的时间尺度与宏观相互作用的时间尺度分开。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所构成的有机体的寿命小几个数量级；同样，单个个体的生或死对该个体所属物种的进化几乎没有影响。然而，当人们想要将此类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会产生巨大的困难，这些方面长期以来阻碍了对此类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仅考虑两个不同的尺度。研究此类系统的一个非常巧妙的想法是假设快速系统演化得如此之快，以至于在慢速系统有机会演化之前就达到了某种平衡。因此，慢速系统将受到快速系统的平均行为的影响，这在某些自然假设下可以很好地理解。这个想法已经在所谓的平均理论中的混沌动力学设置中得到了实现。在我提出的研究计划中，基于我和我的合著者在过去几年中获得的平均理论的一些最新重要成果，我想深入探索此类系统。这将在很大程度上帮助我们理解混沌特性在这个框架中是如何产生和表现的。