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

在许多感兴趣的现实世界系统中，无序是局部产生的，由于与相邻区域的短距离和长距离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，工业过程中的化学品模式，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。 我们量化这些空间相互作用的整体动态的影响，使用两个模型系统，包含基本的物理，同时保持计算访问。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型系统的动力学的更深入的理解。这些发现将为描述社会感兴趣的复杂动力系统所需的理论思想的发展提供见解。该项目包括为初中学生开发一个动手的数值研讨会，为本科生研究提供机会，并支持博士生的研究生研究。 研究结果和最先进的计算方法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本数值研究，具有一系列空间相互作用，强烈驱动平衡。 我们专注于两个模型系统，包含基本的非线性和空间相互作用，同时保持计算可访问，使用动力系统理论的强大思想的基础和广泛的研究。我们将探索离散时间映射的大格和形成称为广义Swift-Hohenberg方程的偏微分方程的正则模式。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的将产生一个前所未有的和定量的描述切线空间动力学。我们将量化的动态双曲的程度，估计惯性流形的尺寸，探索的泛化和鲁棒性的切空间分裂成物理和瞬态模式，并定量地链接的模式动态与时空动态的CLV的。我们将建立一个物理的理解，这些发现是如何变化的功能，强度和长度的空间相互作用，包括在模型系统中。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.