Reference number : GC-2017-Q1-03382, amendment number : , agreement type : G, recipient legal name : De Simoi, Jacopo (University of Toronto), recipient country : CA, recipient province : nan, recipient city : Ontario, recipient postal code : nan, program name english : Discovery Grants Program - Individual, program name french : Programme de subventions à la découverte - individuelles, program purpose english : 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., program purpose french : 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., agreement title english : Fast–slow dynamical systems, agreement title french : Fast–slow dynamical systems, agreement number : RGPIN, agreement value : 125000.0, agreement start date : 2017-05-10, agreement end date : nan, description english : nan, description french : nan, expected results english : nan, expected results french : nan, additional information english : Grant or Award spanning more than one fiscal year. (2017-2018 to 2022-2023), additional information french : Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023), owner organization : nserc-crsng, owner organization title : Natural Sciences and Engineering Research Council of Canada | Conseil de recherches en sciences naturelles et en génie du Canada