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Transfer operators and emergent dynamics in hyperbolic systems

双曲系统中的传递算子和涌现动力学

基本信息

批准号：
EP/V053493/1
负责人：
Wael Bahsoun
金额：
$ 49.75万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV053493%2F1
关键词：
Transfer operators emergent dynamics hyperbolic

项目摘要

Dynamical systems is a field of mathematics that is concerned with studying phenomena that evolve in time. It has deep connections with other areas of mathematics such as analysis, number theory, probability and geometry. Many interesting dynamical systems are chaotic in nature. This means they exhibit sensitive dependence on initial conditions and their long-term behaviour cannot be predicted by following their orbits. Thus, it is natural to study the behaviour of such systems form a probabilistic point of view. An instrumental tool to infer statistical aspects of a chaotic dynamical system is called the transfer operator. Such an operator describes how distributions change over time under the evolution of the dynamics. More importantly, its spectral data encode remarkable information, such as rates of correlation decay, extremes and rare events, on the statistics of the underlying dynamics. A fundamental class of chaotic smooth hyperbolic dynamical systems is called Anosov. In the past fifteen years transfer operator techniques produced impressive results on the statistical aspects of smooth, 'idealised' (single site), Anosov systems and made remarkable new connections with other areas of mathematics, namely with semiclassical analysis, a modern topic in mathematical analysis. However, transfer operator techniques are not yet pioneered for coupled Anosov systems, and hence obviously, not for the more general coupled piecewise hyperbolic systems with singularities. Such coupled systems appear naturally as network models in engineering, physical and biological sciences, and are of paramount importance in studying nonequilibrium thermodynamics. This leaves the fruitful approach of transfer operators, and ergodic theory in general, short on providing statistical insights on the behaviour of such complex systems that are capable of producing emergent dynamics: dynamical quantities, such as escape of mass and heat transfer, that appear as a result of interaction among components in a large system. In this project we aim to achieve a new state-of-the art in smooth ergodic theory and hyperbolic dynamics by developing novel transfer operator techniques to understand emergent dynamical quantities and macroscopic statistical properties of 'large dynamical systems' whose microscopic dynamics are piecewise hyperbolic systems with singularities.

动力学系统是一个数学领域，它关注于研究随时间演化的现象。它与数学的其他领域有着深刻的联系，如分析，数论，概率和几何。许多有趣的动力系统本质上是混沌的。这意味着它们对初始条件表现出敏感的依赖性，并且它们的长期行为无法通过跟踪它们的轨道来预测。因此，从概率的角度来研究这种系统的行为是很自然的。一种用来推断混沌动力系统的统计特性的工具叫做转移算子。这种算子描述了在动态演化下分布如何随时间变化。更重要的是，它的光谱数据编码显着的信息，如相关衰减率，极端和罕见的事件，对潜在的动态统计。混沌光滑双曲动力系统的一个基本类别被称为Anosov。在过去的15年中转让运营商技术产生了令人印象深刻的结果统计方面的顺利，“理想化”（单网站），Anosov系统，并取得了显着的新的联系与其他领域的数学，即与半经典分析，一个现代主题的数学分析。然而，转移算子技术尚未开创耦合Anosov系统，因此显然，不为更一般的耦合分段双曲系统的奇异性。这样的耦合系统自然出现在工程，物理和生物科学中的网络模型，并在研究非平衡态热力学中至关重要。这使得富有成效的方法的传输运营商，遍历理论一般，提供统计的见解，这种复杂的系统的行为，能够产生紧急动态：动态量，如逃逸的质量和热量转移，出现作为一个大系统中的组件之间的相互作用的结果。在这个项目中，我们的目标是实现一个新的国家的最先进的光滑遍历理论和双曲动力学通过开发新的传输算子技术，以了解新兴的动力学量和宏观统计性质的“大动力系统”，其微观动力学是分段双曲系统的奇异性。