Large effects in dynamical systems

这项研究的目的是分析数学机制，解释如何施加在动力系统上的小动作可以导致大的影响，以及这些机制在现实世界中的应用系统，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，由月球、太阳和行星的自然引力交织而产生的微小力，可以用来设计航天器到太阳系内遥远位置的轨道，或者修正地球附近的人造卫星的轨道，从而节省大量燃料。在气候科学中，系统参数的微小变化，如温室气体浓度的变化，可导致气候从一种状态急剧过渡到另一种状态，对气候数据的数学分析可发现这种过渡的早期预警信号。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具有显著的功能效应;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。 该项目旨在发现在各种现实世界系统的一般数学模型中产生巨大影响的显式机制，以及直接从数据中推断这种机制。数据通常用于复杂系统的情况下，其中很难从基本原理中构想出明确的模型。第一个研究方向是致力于大的影响，在机械哈密顿系统受到小扰动，确定性或随机。我们的目标是描述机制，产生的轨迹，在相空间中旅行了很大的距离，以及规定的行程轨迹。 这些机制的应用包括天体动力学和天体力学。第二个研究方向是致力于复杂系统中的关键转变。其目标是开发基于持久同源性的方法，以检测具有缓慢演变参数的嘈杂动力学系统中的关键转变，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GroEL/ES分子伴侣系统参与蛋白质折叠。

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.