Chaos with Multiple Positive Lyapunov Exponents

混沌动力学近年来一直是数学研究的沃土，现在许多科学领域都观察到了混沌现象。到目前为止，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。即使当考虑具有高维状态空间的系统时，动力学通常发生在低维吸引子上(我们所说的低维吸引子，我们指的是映射高达2维，流高达3维)。这项提议试图探索与混沌动力系统相关的一些现象如何扩展到根本上是高维的情况--即当系统在不止一个维度上膨胀时，或者换句话说，当系统具有不止一个正的李亚普诺夫指数时。更高维度的混沌可能出现在许多工程设备的模型中，以及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计算机模拟的可靠性。该研究对复杂系统的真实感建模具有重要意义。

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.