A description of surface dynamics

该项目旨在了解动力系统从有序状态到随机（混沌）状态转变的机制。换句话说，其目的是了解系统的行为从周期性演变为混沌的过程，因为一个或多个控制参数发生变化。一个相关的目标是确定主要的分岔负责的定性变化所表现出的动力系统。虽然这种理解以前已经实现了低维动力系统，这个项目介绍了一种新的方法来超越低维限制。该项目将提供新的概念和方法，为数学和科学的进步提供新的视角。此外，该项目将促进直接从事研究的研究生的培训，并将通过组织暑期学校向本科生提供教育机会，介绍数学方面的主题，包括与动力系统相关的主题。一个理论-三维动力系统成功地解释了混沌现象的深度和复杂性，并描述了典型的典型地图的轨道。它的显着的普遍性属性补充了这种理解与强大的几何工具。在二维环境中，可能出现的动力学情景的范围目前仅被部分理解，而对于那些在一维动力学中不会出现的新现象，仍有待于发展一个总体框架。在先前的工作由美国国家科学基金会的支持，主要研究者介绍了一个大的开放类的二维动力系统，包括经典的Henon家庭没有大面积收缩的限制，这是服从于获得的结果，在一维的情况下。此外，利用重整化方案理解从零熵到正熵的转变也取得了重大进展。本项目有几个组成部分。首先，现有的重整化方案将适应于正熵领域。接下来，将讨论在更一般的情况下，对耗散的仿射的表征的初步步骤。最后，首席研究员将寻求发展的理论，可微重整化没有一个先验假设接近一维设置。这些结果将打开一扇大门，一个全球性的描述耗散同构和他们的行为扰动，带来新的工具和新的观点，顺利动力系统theory.This奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。

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.