Dynamics and Kinetics

该项目解决了复杂系统（即具有复杂动力学的系统和/或具有大量相互作用组件（网络）的系统）的演变分析和预测中的几个核心问题。现实世界和工程网络通常有成千上万个组件。由首席研究员开发的理论允许人们将任何类型的网络压缩成更小的网络，同时保留有关初始大网络的重要信息。在拟建的项目中，这一理论将得到进一步的发展。并与疾病控制中心（CDC）的研究人员合作，应用于对丙型肝炎传播和交叉免疫反应网络的分析。交叉免疫反应意味着免疫系统产生的对抗某些特定类型病毒的抗体有时也会对抗其他类型的病毒。在丙型肝炎、流感和登革热中发现了交叉免疫反应性，这是作者与CDC同事建立的数学模型的基础。关于复杂动力学系统演化的传统观点认为，只有对这类系统进行长期预测才是可能的，因为人们应该等到系统稳定下来。由首席研究员提出的一种新方法，允许人们对混沌系统的演化做出短期预测。它将能够回答诸如“系统最可能的状态是什么？”之类的问题。本研究将这种方法扩展到更大的一类具有复杂动力学的系统，并解决了递归问题。混沌的基本机制的理解，称为散焦，将从本质上扩展。首席研究员对这一机制的发现已经在理论和实验物理学中得到了许多应用。具有各种混沌行为的系统的新的视觉模型可以包含在复杂系统和混沌的基础和高级课程中。提出的研究解决了动力系统和统计力学理论中一些长期存在和技术上具有挑战性的问题，以及新的自然问题。由首席研究员开创的动态和随机系统的有限时间定性性质的新领域将得到进一步发展。这将利用符号动力学，离散数学和概率论的方法和思想的组合。网络的等谱变换理论将进一步发展，特别强调分析网络空间中这些变换序列的吸引子。理解离焦机制，双曲（混沌）的基本机制，将被推进。这使用了一种新的混沌台球，其中聚焦组件不会被迫彼此相距很远。经典的Ehrenfests气体将被显示出意想不到的特性，这要归功于一项新的观察，即一个点和一个有限大小的粒子的台球可以有完全不同的动力学。

The project addresses several central problems in the analysis and prediction of the evolution of complex systems, i.e. the systems with complex dynamics and/or systems with a large number of interacting components (networks). Real world and engineering networks often have tens of thousands of components. Theory, developed by the principal investigator, allows one to compress networks of any type into much smaller networks while preserving important information about the initial large network. In the proposed project, this theory will be further developed. And, in collaboration with Center for Disease Control (CDC) researchers, applied to an analysis of transmission and cross-immunoreactivity networks for Hepatitis C. Cross-immunoreactivity means that antibodies produced by the immune system to fight some specific type of viruses sometimes fights other types of viruses as well. Cross-immunoreactivity is found for Hepatitis C, influenza, dengue, and is a basis for a mathematical model developed by the author with CDC colleagues. Traditional views on evolution of systems with complex dynamics suggest that only long term predictions for such systems are possible, since one should wait until the system stabilizes. A new approach, proposed by the principal investigator, allows one to make short term predictions on the evolution of chaotic systems. It will be possible to answer questions such as, "which state of the system is most likely?". This research will extend this approach to a larger class of systems with complex dynamics and also address problem of recurrence. Understanding of a fundamental mechanism of chaos, called defocusing, will be essentially extended. Discovery of this mechanism by the principal investigator has already found numerous applications in theoretical and experimental physics. New visual models of systems with various types of chaotic behavior can be included into basic and advanced courses on complex systems and chaos.The proposed research addresses some long standing and technically challenging, as well as new natural problems, in the theory of dynamical systems and statistical mechanics. A new area of finite time qualitative properties of dynamical and stochastic systems, pioneered by the principal investigator, will be further developed. This will make use of combinations of methods and ideas of symbolic dynamics, discrete mathematics and probability. The theory of isospectral transformation of networks will be further developed with a special emphasis on analysis of attractors of sequences of such transformations in the space of networks. Understanding the mechanism of defocusing, a fundamental mechanism of hyperbolicity (chaos), will be advanced. This uses new classes of chaotic billiards where focusing components are not forced to be far from each other. Classical Ehrenfests' gas will be shown to have unexpected properties thanks to a new observation that billiards with a point and with a finite size particle can have quite different dynamics.