Dynamics in Two Complex Variables

两个复杂变量的动力学

• 批准号: 0302357
• 负责人: John Smillie
• 金额: --
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0302357&HistoricalAwards=false
• 关键词: Dynamics; Two; Complex; Variables

PI: John Smillie, Cornell UniversityDMS-0302357AbstractThe study of mathematical models that describe how systems change with time continues to provide fundamental mathematical challenges. When the models are nonlinear the long-term behavior, or dynamics, of these models can be chaotic and the dependence of the dynamics on the parameters of the model can be extraordinarily delicate. One classic approach to understanding the fundamental mechanisms that explain nonlinear dynamics is to look for generic behavior in the collection of all maps or diffeomorphisms; another approach is to look carefully at well chosen special families. Recent advances in one variable dynamics show that these approaches can be complementary. In particular the study of the complex quadratic family has led to new results about analytic and smooth unimodal families of real maps. The PI will study a particular family of two-dimensional dynamical systems: polynomial diffeomorphisms in two complex dimensions. These will be studied from many points of view. The PI will use ideas from the study of smooth diffeomorphisms of surfaces, methods from potential theory, methods from dynamics in one complex variable and computer tools. The hope is to provide a pathway for the migration of ideas from one-dimensional complex dynamics to higher dimensional real and complex dynamics.The understanding of mathematical models that describe how systems evolve with time continues to provide essential insights in many areas of science. Such models are used to describe the rhythms of the heart, the pulsing of lasers and the spread of disease. Despite the progress that has been made there are still fundamental problems remaining in the study of such mathematical models. Even seemingly simple models can present formidable mathematical difficulties. In the past 20 years important progress has been made in using ideas connected with fractal objects such as Julia sets and the Mandelbrot set in understanding the dynamics of systems with one degree of freedom. The aim of the proposed research is to bring some of these new methods and concepts to bear on the problem of understanding systems with two degrees of freedom. If these ideas prove fruitful there could be a positive effect on the field of dynamical systems. This could in turn have an influence on a range of scientific fields in which mathematical models of time evolution play a role.

主要研究者：John Smillie，Cornell UniversityDMS-0302357摘要描述系统如何随时间变化的数学模型的研究继续提供基本的数学挑战。当模型为非线性时，这些模型的长期行为或动态可能是混沌的，并且动态对模型参数的依赖性可能非常微妙。理解解释非线性动力学的基本机制的一个经典方法是在所有映射或同构的集合中寻找一般行为;另一种方法是仔细研究精心选择的特殊族。单变量动力学的最新进展表明，这些方法可以是互补的。特别是对复二次映射族的研究，得到了关于真实的映射的解析单峰族和光滑单峰族的新结果。PI将研究一个特殊的二维动力系统族：在两个复杂的维度上的多项式同构。将从多个角度对这些问题进行研究。PI将使用的想法，从研究光滑的曲面，方法从潜在的理论，方法从动态在一个复杂的变量和计算机工具。我们希望为从一维复杂动力学到更高维的真实的复杂动力学的思想迁移提供一条途径。对描述系统如何随时间演化的数学模型的理解继续为许多科学领域提供重要的见解。 这种模型被用来描述心脏的节律、激光的脉冲和疾病的传播。尽管已经取得了进展，仍然有一些基本问题仍然存在于这种数学模型的研究。即使是看似简单的模型，也可能带来难以克服的数学困难。在过去的20年里，在使用与分形对象有关的思想，如Julia集和Mandelbrot集，理解单自由度系统的动力学方面取得了重要进展。提出的研究的目的是把这些新的方法和概念，以承担的问题，理解系统的两个自由度。如果这些想法被证明是富有成效的，那么可能会对动力系统领域产生积极的影响。这反过来可能会对一系列科学领域产生影响，其中时间演化的数学模型发挥了作用。