Fast-slow dynamical systems

快-慢动力系统

• 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

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。