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测的巨大差异(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Lorenz的例子中，全球天气模式由Lorenz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。这项工作开启了一章的研究，这是重要的从业者，以及理论动态学家和计算机科学家。我的提案列出了进一步研究这些具有挑战性的问题的计划。

The main theme of my research is the study of complex behaviour generated by simple 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.