Numerical Analysis of Qualitative Dynamics in Flows

Kalies9973331研究人员开发了确定常微分方程解的定性动力学行为的数值方法。对于这些方法，输入只需要一个微分方程组，相空间的一个区域，而不需要动力学的先验知识。其输出是不变量集到包含非平凡动态的区域的莫尔斯分解。关于这些区域的行为的信息以Conley指数的形式给出，并且可以检测到Morse集之间的连接轨道。这种算法作为研究信息很少的微分方程组的第一步是有益的，特别是在数值模拟解难以可视化的更高维中。这里设想的计算类型提供了对动力系统的某种粗略、定性的描述。然而，来自动力系统理论的工具Conley指数是一种代数拓扑指数，它也允许开发算法来严格证明特定动力学行为的存在性。因此，可靠的算法既可以用作分析不熟悉的系统的工具，也可以用作获得特定动力学的计算机辅助证明的工具；两者都将有广泛的应用。这些算法在计算拓扑学中也有潜在的应用，特别是计算细胞复合体的同调，以及与网格生成和Delaunay三角剖分的联系。微分方程组被用来模拟物理科学中的许多现象：机械系统、化学反应和生物系统。理解这种固有的非线性系统的动力学是全球分析中的一个问题。当底层空间的维度高于三维时，对这类系统的研究往往超出了当前分析技术的范围。计算机能够对这类方程的解进行许多计算，但大多数现有的数值方法不能很好地从高维系统中提取渐近动力学行为。一个问题是，计算机可以生成大量难以分析的数据；通常需要进行图形分析，这在更高的维度上是不切实际的，可能会错过不稳定的对象。这个项目的目的是设计和实现算法来研究高维微分方程渐近动力学。该算法的输出是一个代数索引(由C.Conley设计)，它可以被认为是从计算数据中提取基本动力学信息(包括不稳定对象)的工具，并且不需要理解可视化。

Kalies9973331The investigator develops numerical methods for determining the qualitative dynamical behavior of solutions to ordinary differential equations. For these methods, input requires only a system of differential equations, a region of the phase space, and no a priori knowledge of the dynamics. The output is a Morse decomposition of the invariant set into regions containing nontrivial dynamics. Information about the behavior in these regions is given in terms of the Conley index, and connecting orbits between Morse sets can be detected. Such an algorithm is beneficial as a first step in the study of systems of differential equations for which little information is known, particularly in higher dimensions where visualization of numerically simulated solutions is difficult. The type of computation envisioned here provides a somewhat coarse, qualitative description of a dynamical system. However, the Conley index, a tool from dynamical systems theory, is an algebraic topological index, which also allows for the development of algorithms for rigorously proving the existence of specific dynamical behavior. Therefore, a reliable algorithm can be used both as a tool for analyzing unfamiliar systems and as a tool for obtaining computer-assisted proofs of specific dynamics; both would have a wide variety of applications. The algorithms developed here also have potential applications in computational topology, in particular the computation of the homology of cellular complexes, and connections to mesh generation and Delaunay triangulations.Differential equations are used to model many phenomena in the physical sciences: mechanical systems, chemical reactions, and biological systems. Understanding the dynamics of such inherently nonlinear systems is a problem in global analysis. When the underlying space has dimension higher than three, the study of such systems is often beyond the scope of current analytic techniques. Computers are capable of performing many calculations of solutions to such equations, but most current numerical methods are not well suited for extracting asymptotic dynamical behavior from systems in higher dimensions. One problem is that computers can generate a large quantity of data that is difficult to analyze; often a graphical analysis is performed, which is impractical in higher dimensions and can miss unstable objects. The purpose of this project is to design and implement algorithms to study the asymptotic dynamics of differential equations in higher dimensions. The output of the algorithm is an algebraic index (designed by C. Conley) that could be thought of as a tool for extracting the essential dynamical information (including unstable objects) from computational data and that does not require visualization to be understood.