Chaos with Multiple Positive Lyapunov Exponents

具有多个正李亚普诺夫指数的混沌

• 批准号: 9870183
• 负责人: James Yorke
• 金额: $ 33万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9870183&HistoricalAwards=false
• 关键词: Chaos; Multiple; Positive; Lyapunov; Exponents

9870183YorkeChaotic dynamics has been a fertile area of mathematical research in recent years, and chaos has now been observed in many scientific fields. Much of the progress to date in the study of chaotic dynamics has focused on low-dimensional systems, and many phenomena are well understood in certain low-dimensional cases. Even when systems with higher-dimensional state spaces are considered, often the dynamics takes place on a low-dimensional attractor (by "low", we mean up to dimension 2 for maps and 3 for flows). This proposal seeks to explore how some of the phenomena associated with chaotic dynamical systems extend to cases that are fundamentally higher-dimensional -- that is, when the system is expanding in more than one dimension, or in otherwords has more than one positive Lyapunov exponent.Higher-dimensional chaos can occur in models of many engineering devices and in meteorological models and climate models. This research will tell us, for example, 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 computer simulations for higher dimensional models. This study is critical to realistic modeling of complex systems.

近年来，混沌动力学一直是数学研究的一个丰富领域，现在在许多科学领域都观察到了混沌。迄今为止，混沌动力学研究的大部分进展都集中在低维系统上，许多现象在某些低维情况下得到了很好的理解。即使考虑到具有高维状态空间的系统，动态也经常发生在低维吸引子上（这里的“低”指的是地图的2维和流的3维）。本提案旨在探索与混沌动力系统相关的一些现象如何扩展到基本高维的情况-即当系统在不止一个维度上扩展时，或者换句话说，具有不止一个正李雅普诺夫指数。高维混沌可以出现在许多工程装置的模型中，也可以出现在气象模型和气候模型中。例如，这项研究将告诉我们，数学模型是否会突然被困在常规行为的人工窗口中。这些窗户将是人造的，因为任何实际水平的噪音都会破坏正常的行为。我们的研究还将阐明计算机模拟高维模型的可靠性。该研究对复杂系统的真实建模具有重要意义。