Mathematical Sciences: Numerical Analysis and Computation ofInvariant Manifolds

主要研究者进行分析， 计算的数值方法的实现 动力系统和不变曲线的不变流形 地图。 这项工作的一个特点是实施某些 协调变化，这将大大扩展适用性 算法 在大多数应用中，动力系统 取决于一个或多个参数;然后确定 适当的坐标将与路径的概念相结合 以下. 只要歧管变形顺利时，系统 参数的变化，所提出的方法将允许以下 流形计算 一个更长远的目标是研究 数值上不变流形的分叉，特别是 托里的崩溃。 预计， 击穿将在数字上以几个转折为特征 点的离散解分支，并指出， 转折点随着网格尺寸的细化而增加。 的 拟议的研究将指出发展的方向， 不变流形计算的有效算法， 将为动力学研究提供一个有价值的工具 系统. 非线性动力学系统在许多领域都扮演着重要的角色。 应用科学和工程技术领域。 例如，它们用于模拟飞机周围的气流， 潜艇，发动机的振动，电流 网络和心脏中的血液流动。 所有这些系统 取决于可以部分控制或 监测。 一个重要的实际问题是预测 参数的临界值， 动力系统的发生。 这些变化被称为 分叉 例如，一个长期悬而未决的问题是， 来预测从层流到湍流的转变。 类似地，振动结构可以从准周期分叉， 混沌运动;后者是不可预测的，可能导致 结构的崩溃。 一个动态的 系统的特征通常在于其在 不变流形 本项目将开发数值软件 来计算不变流形并通过计算来追踪它们 因为系统参数改变。重要案例如下： 不变环面，对应于准周期（可预测） 运动;圆环面的破裂对应于过渡 混沌（不可预测）运动。 用的是 这项工作，人们将能够计算值的分歧 参数，即，可以预测参数值， 动力系统将发生重大的质的变化。 该算法将在耦合系统上进行测试。 振荡器和振动机械结构，但他们将 适用于其他应用科学领域， 系统用于建模。

The principal investigator undertakes the analysis and implementation of numerical methods for the computation of invariant manifolds of dynamical systems and of invariant curves of maps. A feature of this work is to implement certain coordinate changes that will greatly extend the applicability of the algorithms. In most applications, the dynamical systems depend on one or more parameters; then the determination of suitable coordinates will be combined with the idea of path following. As long as the manifolds deform smoothly when system parameters change, the proposed methods will permit following the manifolds computationally. A longer-term goal is to study bifurcations of invariant manifolds numerically, in particular the breakdown of tori. It is expected that the region of breakdown will be characterized numerically by several turning points of the discrete solution branch, and that the number of turning points increases as the meshsize is refined. The proposed research will point the way towards the development of efficient algorithms for the computations of invariant manifolds, and will provide a valuable tool for the study of dynamical systems. Nonlinear dynamical systems play an important role in many areas of the applied sciences and of engineering and technology. For example, they are used to model flows around airplanes and submarines, vibrations of engines, currents in electrical networks, and blood flow in the heart. All these systems depend on parameters, which can partially be controlled or monitored. A practical problem of main importance is to predict the critical values of parameters, for which qualitative changes of the dynamical systems occur. These changes are called bifurcations. For example, a long-outstanding problem is to predict the transition from laminar to turbulent flow. Similarly, a vibrating structure can bifurcate from quasiperiodic to chaotic motion; the latter is unpredictable and might lead to the breakdown of the structure. The behavior of a dynamical system is often characterized by its properties on an invariant manifold. This project will develop numerical software to compute invariant manifolds and to follow them computationally as system parameters change. Important cases are given by invariant tori, which correspond to quasiperiodic (predictable) motion; breakdown of the tori corresponds to the transition to chaotic (unpredictable) motion. With algorithms developed in this work, one will be able to compute values for bifurcation parameters, i.e., one can predict parameter values for which major qualitative changes of the dynamical system will occur. The algorithms will be tested on systems of coupled oscillators and vibrating mechanical structures, but they will apply to other areas of the applied sciences where dynamical systems are used for modelling.