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 Yorke 该提案研究高维混沌。李亚普诺夫指数衡量动力系统在不同方向上线性膨胀或收缩的平均速率；具有 N 个正 Lyapunov 指数的系统在 N 个线性独立的方向上扩展，因此系统的吸引子应该至少是 N 维的。然而，吸引子和李亚普诺夫指数可能对参数变化非常敏感，导致参数空间中出现系统表现出低维行为的“窗口”。此外，某些系统中正有限时间李亚普诺夫指数的数量随时间的波动导致了关于数值轨迹的阴影特性的重要问题。我们将研究具有多个正李雅普诺夫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常规行为的人工窗口中。这些窗户是人造的，因为任何现实水平的噪音都会扰乱正常的行为。我们的研究还将揭示高维模型数值轨迹的可靠性。这项研究对于现实建模至关重要。