Dynamics and Kinetics

动力学和动力学

• 批准号: 1600568
• 负责人: Leonid Bunimovich
• 金额: $ 37.5万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600568&HistoricalAwards=false
• 关键词: Dynamics; Kinetics

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.

该项目解决了复杂系统演化分析和预测中的几个核心问题，即具有复杂动态的系统和/或具有大量相互作用组件（网络）的系统。 真实的世界和工程网络通常有数以万计的组件。该理论允许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型网络的重要信息。 在本项目中，这一理论将得到进一步发展。 并且，与疾病控制中心（CDC）的研究人员合作，应用于丙型肝炎传播和交叉免疫反应网络的分析。 交叉免疫反应性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。 交叉免疫反应性被发现用于丙型肝炎、流感、登革热，并且是作者与CDC同事开发的数学模型的基础。 传统的观点认为，复杂动态系统的演化，只有长期的预测，这样的系统是可能的，因为应该等到系统稳定。 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变进行短期预测。 它将有可能回答诸如“系统的哪种状态最有可能？". 这项研究将这种方法扩展到一个更大的一类系统的复杂动态，也解决了问题的复发。 理解的基本机制的混乱，所谓的散焦，将从根本上扩展。主要研究者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。具有各种类型混沌行为的系统的新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。拟议的研究解决了动力系统和统计力学理论中一些长期存在的技术挑战以及新的自然问题。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，特别强调在网络空间中分析这种变换的序列的吸引子。 理解散焦的机制，双曲（混沌）的基本机制，将被推进。 这使用了新的混沌台球类别，其中聚焦组件不会被迫远离彼此。 经典的埃克塞特气体将被证明具有意想不到的性质，这要归功于一项新的观察，即具有一个点和有限大小粒子的台球可以具有完全不同的动力学。