Small-scale structures in dynamical systems: Accurate numerics and renormalization

这是一个数学物理和应用数学研究的建议，重点是在动力系统中的小尺度结构。对小尺度结构的研究是重要的，因为它们与系统的渐近行为有着不可分割的关系——这是动力系统理论的一个中心主题，具有许多重要的应用。我们计划从数值和理论上研究不同过渡到复杂行为的缩放指数之间的全局组织——这是最近在PI及其合作者的数值研究中注意到的一个新现象。我们提出了一个基于重整化群描述的理论解释。虽然大多数关于重整化在动力系统中的应用的研究都集中在重整化算子在特定不动点周围的渐近行为上，但我们相信PI及其合作者提出的方法将导致对重整化算子的全局行为的新见解。我们建议进行的其他项目与准周期强迫系统（特别是具有移动边界的谐振器中的经典和量子效应）以及一些物理系统和临界动力学对象的小尺度特性有关。这是一个数学物理和应用数学研究的建议，重点是在动力系统中的小尺度结构。所提出的研究将加深我们对重整化算子动力学的理解，并可能在重整化群描述发挥重要作用的领域——动力系统、统计物理、流体动力学、场论、固体物理——激发新的研究。该项目将进一步加深我们对一些物理现象和关键动力系统的一般性质的认识。该项目是真正的跨学科——它将采用应用分析方法、高精度数值计算技术和理论物理方法，并将刺激不同数学和科学领域专家之间的互动。它将涉及数学和物理专业的本科生和研究生，为他们提供获得在学术内外的职业道路上有用的知识和技能的机会。

This is a proposal for research in mathematical physics and applied mathematics that focuses on small-scale structures in dynamical systems. Investigations of small-scale structures are important because they are inextricably related to the asymptotic behavior of the system -- a central theme in the theory of dynamical systems with numerous important applications. We plan to investigate numerically and theoretically the global organization among the scaling exponents for different transitions to complex behavior -- a new phenomenon noticed recently in the numerical studies of the PI and his collaborators. We propose a theoretical explanation of some of these phenomena that is based on the renormalization group description. While most research on applications of renormalization to dynamical systems focuses on the asymptotic behavior of the renormalization operator around a particular fixed point, we believe that the methodology proposed by the PI and his collaborators will lead to new insights about the global behavior of the renormalization operator. Other projects that we propose to pursue are related to quasiperiodically forced systems (in particular, the classical and quantum effects in resonators with moving boundaries) and small-scale properties of some physical systems and critical dynamical objects.This is a proposal for research in mathematical physics and applied mathematics that focuses on small-scale structures in dynamical systems. The proposed research will deepen our understanding of the dynamics of the renormalization operator and may spur new research in areas where renormalization group description plays an important role -- dynamical systems, statistical physics, fluid dynamics, field theory, solid state physics. The project will further our knowledge about some physical phenomena and the general properties of critical dynamical systems. The project is truly interdisciplinary -- it will employ methods of applied analysis, techniques for high-accuracy numerical computations, and methods of theoretical physics, and will stimulate the interaction between experts in different areas of mathematics and science. It will involve undergraduate and graduate students majoring in mathematics and physics, offering them opportunities to gain knowledge and skills useful in their career paths, whether within or outside of academia.