Mathematical Sciences: "Chaos with Multiple Positive Lyapunov Exponents

数学科学：“具有多个正李雅普诺夫指数的混沌

• 批准号: 9423843
• 负责人: James Yorke
• 金额: $ 28.36万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-05-01 至 1998-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9423843&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Chaos; Multiple; Positive

DMS-9423843 Yorke This proposal studies higher dimensional chaos. Lyapunov exponents measure the average rates of linear expansion or contraction of a dynamical system in different directions; a system with N positive Lyapunov exponents is expanding in N linearly independent directions, thus an attractor of the system should be at least N-dimensional. However, attractors and Lyapunov exponents can be very sensitive to parameter changes, leading to "windows" in parameter space for which the system exhibits lower-dimensional behavior. Also, the fluctuations over time of the number of positive finite-time Lyapunov exponents in some systems leads to important questions about the shadowing properties of numerical trajectories. We will study these and other fundamental aspects of dynamics systems with multiple positive Lyapunov exponents. Higher dimensional chaos can occur in models of many engineering devices and in meteorological models and climate models. This research will tell us if mathematical models can suddenly get trapped in artificial windows of regular behavior. These windows would be artificial in that any realistic level of noise would disrupt the regular behavior. Our research will also shed light on the reliability of numerical trajectories for higher dimensional models. This study is critical to realistic modeling.

DMS-9423843约克这个提议研究的是更高维度的混沌。李雅普诺夫指数衡量了一个动力系统在不同方向上线性扩张或收缩的平均速率；一个具有N个正李雅普诺夫指数的系统在N个线性无关的方向上扩张，因此该系统的吸引子至少应该是N维的。然而，吸引子和Lyapunov指数对参数的变化非常敏感，导致参数空间中的“窗口”，系统在参数空间表现出低维行为。此外，在一些系统中，正的有限时间Lyapunov指数的数目随时间的波动导致了关于数值轨迹的跟踪性质的重要问题。我们将研究具有多个正Lyapunov指数的动力学系统的这些和其他基本方面。高维混沌可能出现在许多工程设备的模型中，以及气象模型和气候模型中。这项研究将告诉我们，数学模型是否会突然陷入有规律行为的人工窗口。这些窗户是人为的，因为任何逼真的噪音水平都会扰乱正常的行为。我们的研究还将阐明高维模型的数值轨迹的可靠性。这一研究对真实感建模具有重要意义。