Mathematical Sciences: "Chaos with Multiple Positive Lyapunov Exponents

这个建议研究高维混沌。李亚普诺夫指数测量动力系统在不同方向上的线性膨胀或收缩的平均速率；一个有N个正李雅普诺夫指数的系统在N个线性无关的方向上展开，因此系统的吸引子至少应该是N维的。然而，吸引子和Lyapunov指数对参数变化非常敏感，导致系统在参数空间中出现“窗口”，在此窗口中系统表现出较低维的行为。此外，在某些系统中，有限时间李雅普诺夫正指数数随时间的波动导致了数值轨迹的阴影性质的重要问题。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。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。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高维模型的数值轨迹的可靠性。该研究对现实建模至关重要。

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.