Rigidity, Universality, and Complexity in Dynamics

动力学的刚性、普遍性和复杂性

• 批准号: RGPIN-2018-04426
• 负责人: Yampolsky, Michael
• 金额: $ 4.08万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758299
• 关键词: Rigidity; Universality; Complexity; Dynamics

The main theme of my research is the study of complex behaviour generated by simple dynamics. One of the key themes in Dynamics is the study of universal laws which emerge from the complex behaviour of dynamical systems. Such universalities are well-known in physics. They can be understood as a self-organization mechanism, which arises from chaotic behaviour. Perhaps the most famous examples of them are Feigenbaum-type universalities in one-dimensional dynamical systems, which have been intensively studied in the last three decades. These studies founded a new field, known as renormalization theory, which revolutionized the study of dynamics. In a series of papers I completely resolved one of the two main cases of one-dimensional universalities (the Lanford's universality for critical circle maps). Closely related to universality is the phenomenon of rigidity. It postulates that dynamical systems which exhibit the same universal properties are related by a smooth change of coordinates. My proposal outlines a program of study of some of the central questions of universality and rigidity in one-dimensional dynamics.It is a well-known fact that a physical system guided by simple dynamical laws can be difficult to model and predict. A famous example of this is the Lorenz system, which describes a very simplified weather model. Even tiny differences in the initial conditions of the model may lead to a widely diverging weather predictions (a so-called butterfly effect). Nevertheless, it is generally believed that while individual predictions may differ, the overall picture of the dynamics remains the same, and may be simulated numerically. In the Lorenz example, the global weather patterns are described by the Lorenz butterfly attractor, which is a well-defined mathematical object. It is easily computed (indeed, the picture has become famous). In a series of recent works, we have tested the intuitive assumption that the attractor of a simple dynamical system can be modeled on a computer. Surprisingly, for an archetypical family, known as Julia sets, this is generally false. This work opens a chapter of research which is important for practitioners, as well as for theoretical dynamicists and computer scientists. My proposal lays out a plan of further study of these challenging questions.

我的研究主题是研究由简单动力学产生的复杂行为。动力学的关键主题之一是研究从动力系统的复杂行为中产生的普遍规律。这种普遍性在物理学中是众所周知的。它们可以被理解为一种自组织机制，源于混沌行为。也许其中最著名的例子是一维动力系统中的费根鲍姆型普适性，在过去的三十年里，它得到了深入的研究。这些研究建立了一个新的领域，称为重整化理论，它彻底改变了动力学的研究。在一系列的文件，我完全解决了两个主要案件之一的一维普遍性（兰福德的普遍性的关键圆地图）。与普遍性密切相关的是僵化现象。它假定具有相同普适性质的动力系统通过坐标的平滑变化而联系起来。 我的建议概述了一维动力学中普遍性和刚性的一些核心问题的研究计划。众所周知，由简单动力学定律指导的物理系统可能很难建模和预测。一个著名的例子是洛伦兹系统，它描述了一个非常简化的天气模型。即使是模型初始条件的微小差异也可能导致天气预测的巨大差异（所谓的蝴蝶效应）。尽管如此，人们普遍认为，虽然个别预测可能会有所不同，但动态的总体情况保持不变，并且可以进行数值模拟。在洛伦兹例子中，全球天气模式由洛伦兹蝴蝶吸引子描述，这是一个定义明确的数学对象。它很容易计算（事实上，这张照片已经变得很有名）。在最近的一系列工作中，我们已经测试了一个简单的动力系统的吸引子可以在计算机上建模的直观假设。令人惊讶的是，对于一个典型的家庭，被称为朱莉娅集，这通常是错误的。这项工作开辟了一个章节的研究，这是重要的从业人员，以及理论动力学家和计算机科学家。我的建议提出了进一步研究这些具有挑战性的问题的计划。