A description of surface dynamics

表面动力学的描述

• 批准号: 2400008
• 负责人: Enrique Pujals
• 金额: $ 24.91万
• 依托单位: CUNY Graduate School University Center
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2400008&HistoricalAwards=false
• 关键词: description; surface; dynamics

This project seeks to understand the mechanisms that underlie the transition of a dynamical system from an ordered state to a random (chaotic) state. In other words, the aim is to understand the processes through which a system's behavior evolves from periodicity toward chaos, as one or more governing parameters are varied. A related goal is to identify the primary bifurcation responsible for qualitative changes exhibited by a dynamical system. While such comprehension has previously been attained for low-dimensional dynamical systems, this project introduces a novel approach to transcend the low-dimensional limitation. The project will offer new conceptual ideas and approaches to provide fresh perspectives on advances in mathematics and science. Additionally, the project will facilitate the training of graduate students directly engaged in the research, and will afford educational opportunities to undergraduate students through the organization of a summer school presenting topics in mathematics, including topics related to dynamical systems.The theory of one-dimensional dynamical systems successfully explains the depth and complexity of chaotic phenomena in concert with a description of the dynamics of typical orbits for typical maps. Its remarkable universality properties supplement this understanding with powerful geometric tools. In the two-dimensional setting, the range of possible dynamical scenarios that can emerge is at present only partially understood, and a general framework for those new phenomena that do not occur for one-dimensional dynamics remains to be developed. In prior work supported by the NSF, the principal investigator introduced a large open class of two-dimensional dynamical systems, including the classical Henon family without the restriction of large area contraction, that is amenable to obtaining results as in the one-dimensional case. Moreover, major progress was reached to understand the transition from zero entropy to positive entropy using renormalization schemes. The present project has several components. First, existing renormalization schemes will be adapted to the positive entropy realm. Next, initial steps towards a characterization of dissipative diffeomorphisms in more general contexts will be addressed. Finally, the principal investigator will seek to develop the theory of differentiable renormalization without an a priori assumption of proximity to the one-dimensional setting. These results will open the door to a global description of dissipative diffeomorphisms and their behavior under perturbation, bringing both new tools and new perspectives to smooth dynamical systems theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目旨在了解动力系统从有序状态向随机（混沌）状态转变的机制。换句话说，其目的是了解随着一个或多个控制参数的变化，系统行为从周期性演变为混沌的过程。一个相关的目标是确定导致动力系统表现出的质变的主要分叉。虽然以前已经对低维动力系统进行了这种理解，但该项目引入了一种超越低维限制的新方法。该项目将提供新的概念思想和方法，为数学和科学的进步提供新的视角。此外，该项目还将促进直接参与研究的研究生的培训，并将通过组织暑期学校提供数学主题（包括与动力系统相关的主题）的本科生教育机会。一维动力系统理论与典型地图的典型轨道动力学描述相结合，成功地解释了混沌现象的深度和复杂性。其卓越的普适性通过强大的几何工具补充了这种理解。在二维环境中，目前仅部分了解可能出现的动力学场景的范围，并且对于一维动力学不会发生的那些新现象的通用框架仍有待开发。在 NSF 支持的先前工作中，首席研究员引入了一个大型开放类二维动力系统，包括不受大面积收缩限制的经典 Henon 族，可以像一维情况一样获得结果。此外，使用重整化方案在理解从零熵到正熵的转变方面取得了重大进展。目前的项目有几个组成部分。首先，现有的重整化方案将适应正熵领域。接下来，将讨论在更一般的背景下表征耗散微分同胚的初步步骤。最后，主要研究者将寻求发展可微重正化理论，而无需先验地假设接近一维设置。这些结果将为耗散微分同胚及其扰动行为的全球描述打开大门，为平滑动力系统理论带来新工具和新视角。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。