N-point motions in Random Dynamical Systems

随机动力系统中的 N 点运动

• 批准号: 2602126
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 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=studentship-2602126
• 关键词: point; motions; Random; Dynamical; Systems

Random dynamical systems combine classical, deterministic, mathematical models designed to capture the governing essence of a system, i.e. its main driving forces, together with external stochastic fluctuations known as noise, providing us with a significantly more realistic framework to describe a wide range of processes [1]. In order to study these systems, robust analytical tools have been developed from stochastic analysis, ergodic theory and more recently bifurcation theory, allowing not only for a statistical explanation of the model but also for a dynamical path-wise interpretation of its behavior.In this setting, we focus on the study of several (n) particles within a random dynamical system, which we refer to as the n-point motion, to go beyond the usual single-point, statistical, and probabilistic description of a model. Starting with the two-point motion, in the case of stochastic differential equations and particularly for stochastic flows of diffeomorphisms, it was shown by H. Kunita in 1990 [2] that the law of the process is fully characterized by the two-point motion, or in other words that knowledge of the dynamics of any two particles evolving within the system yields a full description of the flow. Indeed, the study of the two-point motion in random dynamical systems is also crucial for the description of synchronization and closely relates to fundamental notions in the field such as Lyapunov exponents or the system's entropy amongst others.More recently, Homburg et al. have observed a close link between the bifurcations on the invariant measure of the two-point motion and phase transitions that provide a much richer understanding of the underlying system and its dynamics. However, a complete theory able to describe this topic is yet to be developed.The aim of this project is to identify and analyze such novel mechanisms, as we access the hidden information behind the two-point motion's dynamics, and continue by building towards a full description of the system's n-point motion, uncovering the properties of such complex models.This project falls within the EPSRC statistics and applied probability, and non-linear systems research areas.

随机动态系统结合了旨在捕获系统的管理本质的经典，确定性的数学模型，即其主要驱动力，以及外部随机波动，称为噪声，为我们提供了一个更现实的框架，以描述广泛的过程[1]。 In order to study these systems, robust analytical tools have been developed from stochastic analysis, ergodic theory and more recently bifurcation theory, allowing not only for a statistical explanation of the model but also for a dynamical path-wise interpretation of its behavior.In this setting, we focus on the study of several (n) particles within a random dynamical system, which we refer to as the n-point motion, to go beyond the usual single-point, statistical, and probabilistic description模型。从两点运动开始，在随机微分方程的情况下，尤其是对于差异性的随机流，H。Kunita在1990年[2]表明，该过程的法律已完全表征了两点运动，或者以两种粒子在系统中的动力学知识在系统中进化的知识产生了整体描述。实际上，对随机动力学系统中两点运动的研究对于描述同步也至关重要，并且与诸如Lyapunov指数或系统熵等领域的基本概念紧密相关。最近，Homburg等人。已经观察到分叉之间在两​​点运动的不变度度量和相变的不变度之间的紧密联系，这些度量为基础系统及其动力学提供了更丰富的理解。但是，一个能够描述该主题的完整理论尚未开发出来。该项目的目的是识别和分析此类新型机制，因为我们访问了两点运动动态背后的隐藏信息，并继续朝着对系统的N点运动进行完整描述，揭示了此类复杂模型的属性。这些项目属于EPSRC Statistic statistical intercrc statistical interiasisation conteriations in epsrc statistical copied Probiencity＆Applied Propied and Nons and and and and and and and and and and and and and and liNlinear and linnlinear。