Complexity and universality in dynamics

我的研究主题是研究由简单动力学产生的复杂行为。众所周知，一个由简单动力学定律指导的物理系统是很难建模和预测的。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的巨大差异（所谓的蝴蝶效应）。然而，人们普遍认为，虽然个别的预测可能会有所不同，但动态的总体情况是相同的，并且可以进行数值模拟。在洛伦兹例子中，全球天气模式是由洛伦兹蝴蝶吸引子描述的，这是一个定义良好的数学对象。这很容易计算（的确，这幅画已经出名了）。在最近的一系列工作中，我们已经测试了一个直观的假设，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，对于一个典型的家庭，即所谓的茱莉亚家庭，这通常是错误的。这项工作开启了对实践者、理论动力学家和计算机科学家都很重要的研究篇章。我的建议为进一步研究这些具有挑战性的问题提出了一个计划。

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.