Complexity and Universality in Dynamics

我研究的主要主题是研究由简单动力学产生的复杂行为。众所周知，由简单的动力学定律指导的物理系统很难建模和预测。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。即使模型初始条件的微小差异也可能导致天气预报大相径庭(即所谓的蝴蝶效应)。然而，人们普遍认为，尽管个别预测可能不同，但动态的整体情况保持不变，并可以用数字模拟。在洛伦茨的例子中，全球天气模式是由洛伦茨蝴蝶吸引子描述的，这是一个定义明确的数学对象。它很容易计算(事实上，这幅画已经出名了)。在最近的一系列工作中，我们验证了一个直观的假设，即一个简单动力系统的吸引子可以在计算机上建模。令人惊讶的是，对于一个被称为朱莉娅·塞茨的典型家庭来说，这通常是错误的。这项工作开启了研究的一个章节，这对实践者以及理论动态学家和计算机科学家都很重要。我的建议列出了进一步研究这些具有挑战性的问题的计划。一个相关的主题是研究从动态系统的复杂行为中出现的普遍规律。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，它源于混乱的行为。也许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，人们对它进行了深入的研究。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。在一系列论文中，我完全解决了一维普适性(临界圆映射的朗福德普适性)的两个主要情况之一。我的提案概述了一维动力学中普适性的研究方案

The main theme of my research is the study of complex behaviour generated by simple dynamics. It is awell-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 tinydifferences in the initial conditions of the model may lead to a widely diverging weather predictions (aso-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-definedmathematical 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 acomputer. Surprisingly, for an archetypical family, known as Julia sets, this is generally false. This work opensa chapter of research which is important for practitioners, as well as for theoretical dynamicists and computerscientists. My proposal lays out a plan of further study of these challenging questions.A related theme is the study of universal laws which emerge from the complex behaviour of dynamicalsystems. Such universalities are well-known in physics. They can be understood as a self-organizationmechanism, which arises from chaotic behaviour. Perhaps the most famous examples of them areFeigenbaum-type universalities in one-dimensional dynamical systems, which have been intensively studied inthe last three decades. These studies founded a new field, known as renormalization theory, whichrevolutionized the study of dynamics. In a series of papers I completely resolved one of the two main cases ofone-dimensional universalities (the Lanford's universality for critical circle maps). My proposal outlines aprogram of study of universality in one-dimensional dynamics