Large effects in dynamical systems

动力系统中的巨大影响

• 批准号: 1515851
• 负责人: Marian Gidea
• 金额: $ 29万
• 依托单位: Yeshiva University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1515851&HistoricalAwards=false
• 关键词: Large; effects; dynamical; systems

The objective of this research is the analysis of mathematical mechanisms that explain how small actions exerted on dynamical systems can lead to large effects, and applications of these mechanisms to real-world systems, such as in astrodynamics, molecular biology and bioengineering, and climate science. In astrodynamics, small forces resulting from the intertwining of the natural gravity of Moon, Sun, and planets, can be exploited to design spacecraft trajectories to distant locations within the Solar System, or to correct the orbits of artificial satellites near the Earth, with significant fuel savings. In climate science, small changes in the system parameters, such as in the concentration of greenhouse gases, can cause sharp transitions of the climate from one regime to another, and the mathematical analysis of climate data can detect early warning signs of such transitions. In molecular biology, small changes in the structure of molecules involved in the process of protein folding can have significant functional effects; such changes can be detected via topological data analysis of electron microscopy imaging. This project seeks to discover explicit mechanisms that produce large effects in general mathematical models of various real-world systems, as well as to infer such mechanisms directly from data. Data is often used in the case of complex systems where explicit models are difficult to conceive from fundamental principles. The first research direction is devoted to large effects in mechanical Hamiltonian systems subject to small perturbations, deterministic or random. The goal is to describe mechanisms that yield trajectories that travel a large distance in the phase space, as well as trajectories with prescribed itineraries. Applications of these mechanisms include astrodynamics and celestial mechanics. The second research direction is devoted to critical transitions in complex systems. The goal is to develop methods based on persistent homology that detect critical transitions in noisy dynamical systems with slowly evolving parameters, and to study topological changes in macromolecular processes. These methods are applied to paleoclimate data and GroEL/ES chaperonin systems involved in protein folding.

这项研究的目的是分析数学机制，解释对动力系统施加的小动作如何导致大的影响，并将这些机制应用于现实世界的系统，例如天体动力学、分子生物学和生物工程以及气候科学。在天体动力学中，可以利用月球、太阳和行星的自然引力相互交织产生的小力来设计前往太阳系内遥远位置的航天器轨迹，或者纠正地球附近的人造卫星的轨道，从而显着节省燃料。在气候科学中，系统参数的微小变化（例如温室气体浓度）可能会导致气候从一种状态急剧转变为另一种状态，而气候数据的数学分析可以检测到这种转变的早期预警信号。在分子生物学中，参与蛋白质折叠过程的分子结构的微小变化可能会产生显着的功能影响；这种变化可以通过电子显微镜成像的拓扑数据分析来检测。 该项目旨在发现对各种现实世界系统的通用数学模型产生巨大影响的显式机制，并直接从数据中推断出此类机制。数据通常用于复杂系统的情况，其中很难根据基本原则构思明确的模型。第一个研究方向致力于机械哈密顿系统中受到小扰动（确定性或随机性）的大影响。目标是描述产生在相空间中行进很长距离的轨迹的机制，以及具有规定行程的轨迹。 这些机制的应用包括天体动力学和天体力学。第二个研究方向致力于复杂系统中的关键转变。目标是开发基于持久同源性的方法，检测参数缓慢演化的噪声动力系统中的关键转变，并研究大分子过程中的拓扑变化。这些方法适用于古气候数据和参与蛋白质折叠的 GroEL/ES 伴侣系统。