The Complex Dynamics of Large Systems with Long-Range Interactions: New Insights from Covariant Lyapunov Vectors

在许多现实世界的系统中，混乱是局部产生的，由于与邻近区域的短期和长期空间相互作用而导致复杂的动力学。例子包括大气和海洋的动力学，工业过程中化学物质的模式，大型分布式网络（如互联网）的动力学，以及大量神经元的非线性相互作用。我们使用两个模型系统来量化这些空间相互作用对整体动力学的影响，这两个模型系统包含基本物理，同时保持计算可访问性。我们使用协变李雅普诺夫向量的强大思想，它量化了小扰动的增长或衰减，以建立对具有空间相互作用的大系统动力学的更深层次的理解。这些发现将为描述社会利益的复杂动力系统所需的理论思想的发展提供见解。该项目包括为初中生开发一个动手数字研讨会，为本科生研究提供机会，并支持博士生的研究生研究。研究结果，最先进的计算方法探索，将被用作高级研究生课程的一部分。该项目是大型空间扩展系统动力学的基础数值研究，具有一系列空间相互作用，被强烈地赶出平衡。我们专注于两个模型系统，其中包含基本的非线性和空间相互作用，同时保留计算可访问性，用于使用动力系统理论的强大思想进行基础和广泛的研究。我们将探索离散时间映射的大格和形成偏微分方程的典型模式，称为广义Swift-Hohenberg方程。我们将使用协变李雅普诺夫向量（CLV）的强大思想来获得基本的新见解。CLV将产生切线空间动力学的前所未有的定量描述。我们将量化动力学的双曲度，估计惯性流形的维数，探索切线空间分裂为物理模式和瞬态模式的泛化和鲁棒性，并定量地将模式动力学与CLV的时空动力学联系起来。我们将建立一个物理理解，了解这些发现如何随着模型系统中包含的空间相互作用的强度和长度尺度的函数而变化。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In many real-world systems of interest, disorder is generated locally, leading to complex dynamics as a result of short and long-range spatial interactions with neighboring regions. Examples include the dynamics of the atmosphere and oceans, the patterns of chemicals in industrial processes, the dynamics of large distributed networks such as the internet, and the nonlinear interactions of large numbers of neurons. We quantify the impact of these spatial interactions on the overall dynamics using two model systems that contain the essential physics while remaining computationally accessible. We use the powerful idea of covariant Lyapunov vectors, which quantify the growth or decay of small disturbances, to build a deeper understanding of the dynamics of large systems with spatial interactions. These findings will provide insight for the development of the theoretical ideas needed to describe complex dynamical systems of societal interest. The project includes the development of a hands-on numerical workshop for junior high students, provides opportunities for undergraduate research, and supports the graduate research of a PhD student. The research findings, and the state-of-the-art computational approaches explored, will be used as part of an advanced graduate course.This project is a fundamental numerical investigation of the dynamics of large spatially-extended systems, with a range of spatial interactions, that are strongly driven out of equilibrium. We focus on two model systems that contain the essential nonlinearities and spatial interactions, while remaining computationally accessible, for a fundamental and broad study using powerful ideas from dynamical systems theory. We will explore large lattices of discrete-time maps and a canonical pattern forming partial differential equation called the Generalized Swift-Hohenberg equation. We will use the powerful idea of covariant Lyapunov vectors (CLV's) to gain fundamental new insights. The CLV's will yield an unprecedented and quantitative description of the tangent-space dynamics. We will quantify the degree of hyperbolicity of the dynamics, estimate the dimension of the inertial manifold, explore the generalization and robustness of the tangent-space splitting into physical and transient modes, and quantitatively link the pattern dynamics with the spatiotemporal dynamics of the CLV's. We will build a physical understanding for how these findings vary as a function of the strength and length-scale of the spatial interactions that are included in the model systems.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.