Mathematical Sciences: Numerical Analysis and Computation ofInvariant Manifolds

主要研究者负责分析和实现计算动力系统的不变流形和地图的不变曲线的数值方法。这项工作的一个特点是实现了一定的坐标变化，这将大大扩展算法的适用性。在大多数应用中，动力系统依赖于一个或多个参数；然后结合路径跟踪思想确定合适的坐标。只要流形在系统参数变化时能够平滑变形，所提出的方法就允许对流形进行计算跟踪。一个长期的目标是用数值方法研究不变流形的分岔，特别是环面分解。预计击穿区域将由离散解分支的几个拐点来数值表征，并且随着网格尺寸的细化，拐点的数量会增加。所提出的研究将为开发有效的计算不变流形的算法指明方向，并将为动力系统的研究提供有价值的工具。非线性动力系统在应用科学和工程技术的许多领域发挥着重要作用。例如，它们被用来模拟飞机和潜艇周围的流动、发动机的振动、电网中的电流和心脏中的血液流动。所有这些系统都依赖于参数，这些参数可以部分地控制或监测。一个重要的实际问题是预测参数的临界值，因为这些参数会引起动力系统的质变。这些变化被称为分岔。例如，一个长期悬而未决的问题是如何预测从层流到湍流的过渡。类似地，振动结构可以从准周期运动分岔到混沌运动；后者是不可预测的，可能导致结构的崩溃。动力系统的行为通常由它在不变流形上的性质来表征。该项目将开发数值软件来计算不变流形，并随着系统参数的变化进行计算。重要的情况由不变环面给出，它对应于准周期（可预测）运动；环面分解对应于向混沌（不可预测）运动的过渡。利用在这项工作中开发的算法，人们将能够计算分岔参数的值，也就是说，人们可以预测动力系统将发生重大质的变化的参数值。这些算法将在耦合振荡器和振动机械结构的系统上进行测试，但它们将适用于应用科学的其他领域，其中动力系统用于建模。

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.