Mathematical Sciences: Numerical Analysis and Computation ofInvariant Manifolds

数学科学：不变流形的数值分析与计算

• 批准号: 9107612
• 负责人: Jens Lorenz
• 金额: $ 7.42万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1991
• 资助国家: 美国
• 起止时间: 1991-08-01 至 1995-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9107612&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Numerical; Analysis; Computation

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.

主要研究人员负责动力系统不变流形和映射不变曲线计算的数值方法的分析和实现。这项工作的一个特点是实现了某些坐标变化，这将极大地扩展算法的适用性。在大多数应用中，动力系统依赖于一个或多个参数；然后，确定合适的坐标将与路径跟踪的思想相结合。只要流形在系统参数变化时平稳变形，所提出的方法就可以在计算上跟踪流形。一个较长期的目标是用数值方法研究不变流形的分支，特别是环面的破裂。预计击穿区域将由离散解分支的几个转折点来数值表征，并且转折点的数量随着网格尺寸的细化而增加。所提出的研究将为发展计算不变流形的有效算法指明方向，并将为动力系统的研究提供一个有价值的工具。非线性动力系统在应用科学和工程技术的许多领域都有着重要的作用。例如，它们被用来对飞机和潜艇周围的流动、发动机的振动、电力网络中的电流以及心脏中的血液流动进行建模。所有这些系统都依赖于参数，这些参数可以部分控制或监控。一个重要的实际问题是预测参数的临界值，对于临界值，动力系统会发生质的变化。这些变化被称为分叉。例如，一个长期悬而未决的问题是预测从层流到湍流的转变。类似地，振动结构可以从准周期运动分叉到混沌运动；后者是不可预测的，可能会导致结构的崩溃。动力系统的行为通常由它在不变流形上的性质来刻画。这个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；环面破裂对应于向混沌(不可预测)运动的转变。利用本工作开发的算法，可以计算分叉参数的值，即可以预测动力系统将发生重大质变的参数值。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，但它们将适用于使用动力系统进行建模的其他应用科学领域。