Mathematical Sciences: "Chaos with Multiple Positive Lyapunov Exponents

DMS-9423843 Yorke这个提议研究更高维的混沌。李雅普诺夫指数测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个正李雅普诺夫指数的系统在N个线性独立的方向上膨胀，因此系统的吸引子应该至少是N维的。然而，吸引子和李雅普诺夫指数可以是非常敏感的参数变化，导致在参数空间中的“窗口”，系统表现出低维的行为。此外，随着时间的波动，在某些系统中的正有限时间李雅普诺夫指数的数量导致重要的问题的阴影性质的数值轨迹。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。这些窗口将是人为的，因为任何现实水平的噪音都会扰乱正常的行为。我们的研究也将揭示更高维模型的数值轨迹的可靠性。该研究对真实感建模具有重要意义。

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.